waterheater

Typical residential water heaters range in size from about 30 gallons up to 120 gallons or so, with most tanks falling in the range of about 40 to 80 gallons. Hot water is drawn from the top of the tank to service the home's water loads, with cold make-up water injected near the bottom of the tank. When a hot water draw is in progress, the tank will tend to have a layer of cold water at the bottom.

GridLAB-D™ supports three heating modes (heat_mode): electric resistance (ELECTRIC), natural gas (GASHEAT), and heat pump (HEAT_PUMP). The choice of heating mode determines both which physical model is used and which published properties are relevant during a simulation.

Water is typically heated by two elements, one located near the bottom of the tank and one located higher in the tank, usually halfway to two-thirds of the way to the top. The lower element shoulders most of the heating load, with the upper element engaging only when the cold water layer has reached it. The two elements are controlled by independent thermostats, but only one is permitted to be on at a time, the upper element having priority. The design is intended to allow rapid (re)heating of the smaller volume of water above the top element so that full-temperature water is available as soon as possible after a depletion. Water heater elements range in capacity from about 1500 Watts to 6000 Watts, with 4500 Watts being common.

Thermostatic controls have a "dead band" associated with the setpoint, which prevents rapid cycling of power to the elements which would result if the turn-on temperature equaled the turn-off temperature. The dead band is typically a few degrees above and below the nominal setpoint.

Most heaters are shipped with both the upper and lower thermostats set to the same temperature (often 120 degrees-F), but they may be modified at installation or a later time. Some manufacturers recommend that if hotter water is desired, only the lower element be set to a higher temperature. This will maintain the entire tank at the higher temperature but allow for rapid recovery of the upper volume to the lower setting following a depletion. The net effect on energy use is negligible if the upper thermostat is set to a lower setting. However, some homeowners wrongly set the upper element to a higher temperature than the lower one, which results in somewhat different behavior. Because of thermal stratification, the upper volume will be maintained at a higher temperature than the lower volume, resulting in somewhat lower standby losses than when the entire tank is heated to the higher setting.

The waterheater uses one of four simulation models (waterheater_model): a one-node lumped-parameter model (ONENODE), a two-node stratified model (TWONODE), a physics-based Fortran model (FORTRAN), or a multi-layer discretized model (MULTILAYER). For the standard electric and gas modes the model is selected automatically at runtime based on the current tank state; FORTRAN and MULTILAYER must be selected explicitly by the user.

Default Waterheater

An empty waterheater object, along the lines of

object waterheater { }

will be constructed into a semi-consistent state. The "default waterheater" ends up being similar to

object waterheater {
   tank_volume 50.0 gal;
   tank_diameter 1.5 ft;
   tank_height 3.782 ft;
   inlet_water_temperature 60.0 degF;
   location GARAGE;                  // 80% probability; 20% INSIDE
   heat_mode ELECTRIC;
   waterheater_model NONE;           // auto-selects ONENODE or TWONODE at runtime
   tank_setpoint random.normal(125,5);                  // bound [90, 160]
   thermostat_deadband (1 + fabs(random.normal(2,1)));  // bound [1, 10]
   tank_UA random.normal(2.0, 0.2) * tank_volume/50;    // bound [0.1, 10]
   heating_element_capacity 4.5 kW;  // tanks >= 50 gal; smaller tanks may be 3.2 or 3.5 kW
   re_override OV_NORMAL;
}

Any properties that are not set explicitly will carry these default values.

Waterheater Properties

<table-caption identifier="Table" id=tbl:table-tank }

Operating Mode and Model Selection

Table: Operating Mode and Model Selection { id=tbl:table-oppmode >

Core Tank Properties

Status and Output Properties

Gas Waterheater Properties

Heat Pump Waterheater Properties

FORTRAN Model Properties

These properties are only relevant when waterheater_model FORTRAN. Valid tank sizes are 40 and 80 gallons only.

MULTILAYER Model Properties

These properties are only relevant when waterheater_model MULTILAYER.

Operating Modes

ELECTRIC: Electric Resistance

heat_mode ELECTRIC;

The default heating mode. One or two resistive heating elements immersed in the tank heat the water directly. The thermostat compares temperature to tank_setpoint ± thermostat_deadband/2 and turns the element on or off accordingly. heating_element_capacity defaults to 4.5 kW for tanks ≥ 50 gallons and is randomly chosen from 3.2, 3.5, or 4.5 kW for smaller tanks if not specified.

Heat losses from the element dissipate into the location of the tank (INSIDE or GARAGE). gas_fan_power and gas_standby_power are not used in this mode; they are automatically set to non-negative values but have no physical effect on the electric model.

GASHEAT: Natural Gas

heat_mode GASHEAT;

A natural gas burner heats the water. The thermodynamic tank model (ONENODE/TWONODE) is the same as for ELECTRIC; only the power source differs. Two additional properties capture the auxiliary electrical loads:

• gas_fan_power: Power consumed by the combustion air fan or vent fan while the burner is firing. Defaults to 1% of heating_element_capacity if not set.
• gas_standby_power: Standby electrical load (e.g., electronic pilot or thermostat logic) when the burner is off. Defaults to 0 W if not set.

The heating_element_capacity property represents the gas burner's thermal output in kW, and it obeys the same defaulting rules as for ELECTRIC mode.

HEAT_PUMP: Heat Pump Water Heater

heat_mode HEAT_PUMP;
waterheater_model NONE;   // uses built-in empirical HPWH model (not FORTRAN)

A refrigerant-cycle heat pump extracts heat from the surrounding air and deposits it into the tank water, achieving a COP greater than 1.0. A backup electric resistance element activates when the heat pump cannot keep up with demand.

Empirical energy-tracking model (default HPWH):

GridLAB-D™'s native HEAT_PUMP implementation uses an empirical model based on an internal energy-deficit tracker called energytake (published as heatpump_predicted_energy). energytake represents the amount of energy the heat pump would need to consume to bring the tank back to setpoint temperature.

• Idle mode (Ton ≤ Tw ≤ Toff, element capacity = 0): The heat pump is off. energytake increases logarithmically with time to reflect standby heat losses. Tank temperature is back-calculated from energytake.
• Active mode (Tw < Ton or element capacity > 0): The compressor or electric element is running. energytake decreases as heat is added.
• If energytake > compressor_max_threshold (default 2000 Wh), the electric resistance backup element activates.
• If energytake < heating_element_min_threshold (default 1100 Wh), the backup element deactivates.
• The compressor activates when energytake > compressor_min_threshold (default ≈ thermostat_deadband/2 × 2.44 × tank_volume).

The thermostat in HEAT_PUMP mode uses a blended Tcontrol temperature rather than Tw directly when the tank is in a PARTIAL state.

COP is calculated internally and published via heat_pump_coefficient_of_performance. It is not a user input.

FORTRAN-based HPWH model:

For a higher-fidelity heat pump simulation, set waterheater_model FORTRAN. See the FORTRAN Model section below.

Waterheater Models

The waterheater_model property selects the simulation model. For ELECTRIC and GASHEAT modes the default (NONE) automatically transitions between ONENODE and TWONODE based on the current tank state. FORTRAN and MULTILAYER must be selected explicitly.

ONENODE and TWONODE — Standard Model

The standard model (waterheater_model NONE, ONENODE, or TWONODE) is the default for ELECTRIC and GASHEAT modes and is also used by the native HEAT_PUMP mode.

Modeling Assumptions

The GridLAB-D™ approach to modeling electric water heaters is designed to be computationally fast yet reasonably accurate. It accommodates the common two-element design and the possibility for "inverted" thermostat settings, wherein the upper element maintains a higher temperature than the lower element.

To achieve the necessary computational speed, the standard model makes the following assumptions:

1. Thermal stratification in the tank is not directly modeled. Depending on the situation, the water will be considered to be either of uniform temperature throughout the tank or "lumped" into two temperature regions (hot and cold layers).
2. The injection of cold inlet water at the bottom of the tank results in either complete mixing with the hot water in the tank or no mixing at all, depending on the volumetric flow rate.

The water heater simulation uses two different models depending on the state of the tank at any given moment:

1. One-Node Model – A simple, lumped-parameter electric analogue model that considers the entire tank to be a single "slug" of water at a uniform temperature. This model concerns the temperature of the water at any given time and/or the time required for the temperature to move between two specified points.
2. Two-Node Model – Applies when the heater is in a state of partial depletion. Considers the heater to consist of two slugs of water, each at a uniform temperature. The upper "hot" node is near the heater's setpoint temperature, while the lower "cold" node is near the inlet water temperature. This model concerns the location of the boundary between the hot and cold nodes, calculating the movement of that boundary as hot water is drawn from the tank and/or heat is added to the tank.

The water heater simulation keys on two primary "states" of the water heater:

• Tank State – The tank can be in one of three states (read via current_tank_status):
  • FULL – All the water in the tank is at a uniform temperature near the heater's setpoint. The One-Node model applies.
  • PARTIAL – The tank is in a state of partial depletion, where some of the hot water has been (or is being) drawn out, leaving hot and cold layers of water in the tank. The Two-Node model applies.
  • EMPTY – The tank has been completely depleted; all the water is at a uniform temperature near the water inlet temperature. The One-Node model applies.
• Load State – This refers to the current water load on the heater; that is, whether and how fast hot water is being drawn from the top of the tank (read via load_state). Formally, this load state applies only when the tank state is PARTIAL, but is useful for the FULL and EMPTY tank states because it tells whether the tank will begin to move toward a PARTIAL state or stay in the current state. There are three possible load states:
  • DEPLETING – Hot water is being drawn at a rate sufficient to move the boundary between the hot and cold zones upward. That is, hot water is being drawn out faster than the heating element can warm the incoming cold water, so the upper layer of hot water is getting smaller and the lower layer of cold water is getting larger.
  • RECOVERING – Hot water is either not being drawn from the tank or is being drawn at a low enough level that the boundary between the hot and cold layers is moving downward. That is, the hot water is being drawn out at a rate low enough that the heating element can warm the replacement water faster than it is being introduced, causing the upper layer of hot water to get larger and the lower layer of cold water to get smaller.
  • STABLE – Simplistically, this state implies that hot water is being drawn at a rate that matches the heating element's ability to warm the incoming cold water. Actually, the hot water draw does not have to exactly match the warming rate for this state to apply. Because there are other heat flows acting on the water (e.g., jacket losses to surrounding air), in any given situation where the tank state is PARTIAL there is a range of hot water flow rates for which the tank will never reach either the FULL or the EMPTY state.

For example, if the tank begins near the FULL state and hot water is drawn from it at a rate slightly higher than the heating element can match, the hot/cold boundary will begin to move upward. However, as the boundary moves up, the size of the hot layer gets smaller, reducing the area of warm tank that is exposed to the surrounding air. The lowered area means lowered heat losses, which may tip the balance so that the heating element eventually can keep up with the hot water draw. Similarly, starting with an almost empty tank, the heating element's capacity may exceed a small hot water draw and move the hot/cold boundary downward until jacket losses tip the balance. When the load state is STABLE, the calculated time to transition is infinite.

This STABLE state is illustrated in Figure 1. For an example water heater, the figure shows the time required to deplete an initially full tank ("time to transition") at various hot water flow rates. Note that flow rates above roughly 0.45 gpm result in positive and finite times to transition. Flow rates below about 0.44 gpm are finite and negative, meaning the tank is not depleting at all, but is actually recovering (i.e., the heating element can more than keep up with the heat removed by the water draw). In between those two points—between the positive and negative spikes to infinity—the tank is in the STABLE state that will never reach either the FULL or the EMPTY state.

These two critical flow rates depend on many factors, including the tank size and shape, tank \(UA\), heating element capacity, and the hot and cold layer temperatures. They are identified in the GridLAB-D™ water heater model by calculating the rate of change of the hot/cold boundary position \(h\) with respect to time \((dh/dt)\). Note in the lower graphic of Figure 1 that the time to transition is positive when \(dh/dt\), calculated at the FULL starting point \(h0\), is negative (meaning the tank is depleting). Also note that the time to transition is negative (meaning the tank is really recovering and will remain FULL) when \(dh/dt\), calculated at the target EMPTY state, is positive. The area between the two critical flow rates is identified by differing signs between \(dh/dt\) calculated at FULL and \(dh/dt\) calculated at EMPTY.

Modeling Approach

Figure 2 shows a schematic representation of the water heater model in which Tavg is the average water temperature throughout the tank and Tamb is the ambient temperature. The thermal capacitance of the water Cw is a function of the tank volume:

The thermal conductance of the tank shell (or "jacket") \(UA\) is calculated from the known R-values of the sides and top of the tank divided into their corresponding areas.

One-Node Model

Considering Figure 2 and treating the water heater as a single node with thermal capacitance \(C_w\), a conductance \(UA\) to ambient conditions, with mass flow rate and heat input rate of \(Q_{elec}\), a heat balance on the water node is as follows:

or,

The time required to change the tank's temperature from an initial temperature \(T_0\) to a new temperature \(T_1\) is given by integrating that equation.

That is an integral of the form \(dx/(a+bx)\), which has solution \(\log(a+bx)/b\). Therefore, the final model of the time required to raise (or lower) the tank's temperature is

Where

• \(a = \frac{\dot m C_p T_{inlet} + U A T_{amb} + Q_{elec}}{C_w}\)
• \(b = \frac{ U A + \dot m C_p }{C_w}\)

The reverse problem of calculating the new temperature of the tank from a known initial temperature and time difference, \(t1-t0\), follows directly:

Two-Node Model

This model, which applies when the heater is in a state of partial depletion, considers the heater to consist of two slugs of water, each at a uniform temperature. The upper "hot" node is near the heater's setpoint temperature, while the lower "cold" node is near the inlet water temperature. The time required to change the tank's hot water column from an initial height of \(h_0\) to a final height of \(h_1\) is given by the following equation:

Where \(dh/dt\) is the location of temperature boundary along the height of the water column. This is calculated as a function of mass flow rate and the temperature difference between the upper and lower interface layers of the water column, as given by the following:

where

• \(a = \frac{Q_{elec}+U A T_{amb}}{C_w T_{lower}} - \frac{\dot m C_p}{C_w}\)
• \(b = \frac{U A}{C_w}\)

In the two-node model, during each synchronization cycle, the height of the hot water column is calculated based on the mass flow rate, using the following equation:

where \(cA\) and \(cb\) are the numerically computed coefficients from the discretized dhdt expression evaluated at the upper-node temperature difference.

FORTRAN Model

waterheater_model FORTRAN;

The FORTRAN model uses a high-fidelity physics-based simulation originally written in Fortran. It is restricted to 40-gallon and 80-gallon tanks only; other volumes will cause a simulation error.

For the 80-gallon tank, operating_mode selects the sub-type:

The Fortran model requires fortran_sim_time to be set to a positive value (in seconds) representing the simulation interval per GridLAB-D™ synchronization step. Physical parameters such as tank geometry, element positions, sensor positions, heat loss rate, and COP are determined automatically from lookup tables based on tank_volume and operating_mode.

Key FORTRAN model parameters set automatically by init():

Use dr_signal to send a demand-response on/off command to the Fortran model. The COP and waterheater_power properties provide read-back of the Fortran model's current operating point.

MULTILAYER Model

waterheater_model MULTILAYER;

The MULTILAYER model discretizes the tank vertically into 10 internal layers (12 total states including boundary nodes), providing a higher-resolution temperature profile than the TWONODE approach. It is computationally more expensive and numerical stability is sensitive to the choice of discrete_step_size.

• Thermostat control: Independent setpoints and deadbands are maintained for the upper (layer 10) and lower (layer 1) heating elements:

• Upper element: upper_tank_setpoint ± upper_tank_deadband/2 — has priority; lower element cannot fire while upper element is on.

• Lower element: lower_tank_setpoint ± lower_tank_deadband/2 — fires only when the upper element is off.

Both default to tank_setpoint and thermostat_deadband if not explicitly set.

• Circulation flow: When a heating element is on, a heuristic recirculation flow (circular_flow_rate, default 2 gpm, range 1–3 gpm) is applied to the zone surrounding that element to model mixing in the vicinity of the heater.

• Mixing valve: The T_mixing_valve property represents a mixing valve reference temperature. Water demand is scaled by a mixing fraction such that the delivered water temperature at the tap is approximately T_mixing_valve. If T_mixing_valve is not set, it defaults to Tmin_upper. Setting it above Tmax_upper is allowed but reduces mixing efficiency.

• State read-back: lower_tank_temperature (layer 1) and upper_tank_temperature (layer 10) expose the temperatures at each element's sensor location. lower_heating_element_state and upper_heating_element_state report the current on/off state of each element.

• Initialization constraints:

• discrete_step_size must be in [1, 60] seconds; values outside this range are silently clamped.
• circular_flow_rate must be in [1, 3] gpm; values outside this range are silently clamped.
• T_mixing_valve should be ≤ upper_tank_setpoint for effective mixing.

Demand Response Override

The re_override property provides direct external control over the heating element, bypassing the internal thermostat:

OV_ON does not override the 212°F boiling protection; the element will still be shut off if the tank approaches boiling.

The FORTRAN model uses the separate dr_signal property instead of re_override.

Simulation Sequence

Each time the water heater is sync'd, the simulation follows four steps:

1. Calculate the energy consumed since the last iteration. The heater remembers whether it was heating and simply computes the consumption based on the time interval since the last sync.
2. Update the tank temperature or the location of the hot/cold boundary, depending on whether the tank was previously FULL, PARTIAL, or EMPTY.
3. Discern whether the tank needs heat. If the tank is in (or has reached) a FULL state at the thermostat setting, the power will be turned off. Otherwise the element will be turned on. Note that, for the Single-Node model, the heater state remains unchanged from its previous state when the water temperature is between the heating cut-off and cut-on temperatures (the deadband around the thermostat setpoint).
4. Calculate and post the time to next transition. For example, if the heater is on, this is either the time for the water to reach the cut-off temperature or for the hot/cold boundary to reach the bottom of the tank, depending on the tank state.

For the MULTILAYER model, steps 2–4 are replaced by a discrete-time matrix simulation over a horizon of discrete_step_size seconds per internal step. The model recalculates the full transition horizon whenever operating conditions change (water demand, inlet temperature, setpoints, or override state).

For the FORTRAN model, the external Fortran subroutine is called each synchronization step with the current operating conditions, and the results (waterheater_power, COP) are read back directly.

Complicating Factors

Several factors complicate the simulation. First, switching models between FULL, PARTIAL, and EMPTY states requires careful attention to not only what's happening now, but what state(s) things were in previously. Second, identifying when the load state is STABLE can be difficult. Finally, each of the models (One-Node and Two-Node) has two incarnations—one to calculate time to transition, and a second "inverted" one to calculate temperature/boundary location after a given amount of time. Because of limitations on floating point precision, the inverted model may not show that the water heater has actually reached the state it was anticipating. For example, the time model may show that the tank will reach the cut-off temperature of, say, 135 degrees in 12 minutes. After 12 minutes have elapsed, the core updates the water heater as expected. The temperature model may calculate that the final temperature is in fact 134.95 degrees. In a worst case the simulation could go into an infinite loop and never actually change states.

Waterheater State of Development

Waterheater is considered a stable model, with a fair amount of functionality.