First-Pass TOU Scheduling Decision Model for HPWHs in OCHRE¶

Switchbox 2025-06-23

Introduction¶

This document outlines a simple, first-pass heuristic-based decision model for consumer response to time-of-use (TOU) electricity rates in residential building simulations using OCHRE, focusing on Heat Pump Water Heaters (HPWHs). The goal is to integrate an adaptive control schedule for the HPWH based on feedback from utility bills and anticipated cost savings, while accounting for consumer effort in shifting schedules.

1. Problem Definition¶

This section establishes a high-level overview of the core research question and modeling assumptions for consumer response to TOU rates.

Objective:

Model how a consumer, with fixed (exogenous) hot water usage needs, might reschedule their HPWH operation in response to TOU rates to minimize their electricity bills, considering a “switching cost” for effort/comfort.

Assumptions:

Hot water usage schedule is fixed (not flexible) and set by inputs to the model.
HPWH can be controlled (on/off) on a schedule.
TOU rate structure (on-peak/off-peak) is known and simple (e.g., higher price during peak hours), such that the consumer can reasonably set a schedule to turn on HPWH during off-peak hours.
Consumer receives feedback on energy cost and makes switching decisions at bill receipt each billing cycle (monthly) rather than at each operational time step (every 15 minutes).
Switching to a TOU-adapted schedule incurs a one-time “effort cost” (can be fixed or parameterized).
Decision process is repeated each billing cycle (feedback loop), with the decision model output feeding into the next iteration of the simulation.

2. Key Variables and Parameters¶

This section defines all variables, parameters, and their temporal dimensions used throughout the decision model. Note that at the beginning of each month, the model is initialized with the previous month’s decision outcome, and the model is repeated for each month of the year, or however many months are specified in the simulation.

Symbol	Type	Description	Units	Dimension
Sets
$M$	Set	Months in simulation year, $m \in \{1, 2, ..., 12\}$	-	12 × 1
$T$	Set	Time periods in billing month, $t \in \{1, 2, ..., T\}$ where $T \approx 2976$ (15-min intervals)	-	\|T\| × 1
$H$	Set	Peak hour periods, $H \subset T$	-	\|H\| × 1
Parameters
$U_{m,t}^{HW}$	Parameter	Exogenous hot water usage schedule at time $t$ in month $m$	L/15min	M × T
$r^{on}$	Parameter	TOU electricity rate during peak hours	$/kWh	1 × 1
$r^{off}$	Parameter	TOU electricity rate during off-peak hours	$/kWh	1 × 1
$C^{switch}$	Parameter	Consumer switching/“hassle” cost per schedule change	$	1 × 1
$\alpha$	Parameter	Monetization factor for comfort penalty	$/kWh	1 × 1
$T_{m,t}^{setpoint}$	Parameter	Hot water temperature setpoint at time $t$ in month $m$	°C	M × T
$T_{m,t}^{ambient}$	Parameter	Ambient water temperature at time $t$ in month $m$	°C	M × T
$\rho$	Parameter	Water density	kg/L	1 × 1
$c_p$	Parameter	Specific heat of water	J/kg·°C	1 × 1
$COP$	Parameter	Heat pump coefficient of performance	-	1 × 1
Decision Variables
$x_m^{switch}$	Binary	Decision to switch schedule in month $m$ (1 = switch, 0 = stay)	binary	M × 1
State Variables
$S_m^{current}$	Binary	Current schedule state in month $m$ (1 = default, 0 = TOU-adapted)	binary	M × 1
$s_{m,t}$	Binary	HPWH operation permission at time $t$ in month $m$ (1 = allowed, 0 = restricted)	binary	M × T
$r_{m,t}$	Variable	Electricity rate at time $t$ in month $m$ (determined by peak/off-peak)	$/kWh	M × T
$E_{m,t}$	Variable	Electricity consumption from HPWH operation at time $t$ in month $m$	kWh/15min	M × T
$T_{m,t}^{tank}$	Variable	Tank water temperature at time $t$ in month $m$	°C	M × T
$Q_{m,t}^{unmet}$	Variable	Thermal unmet demand at time $t$ in month $m$	J/15min	M × T
$D_{m,t}^{unmet}$	Variable	Electrical equivalent unmet demand at time $t$ in month $m$	kWh/15min	M × T
Derived Variables
$C_m^{bill}$	Variable	Monthly electricity bill for water heating in month $m$	$	M × 1
$C_m^{comfort}$	Variable	Monthly comfort penalty from unmet demand in month $m$	$	M × 1
$\Delta C_m$	Variable	Realized bill savings from TOU schedule vs. default in month $m$	$	M × 1

3. Detailed Model Steps¶

This section outlines the complete sequential decision-making process that consumers follow each month, including the initialization of all state variables, decision logic, and state transitions. The model operates on monthly billing cycles with two distinct decision contexts based on the consumer’s current schedule state.

Step 1: Initialize Monthly State Variables¶

This step loads the exogenous input data and sets the initial state variables for month $m$’s simulation. The hot water usage profile $U_{m,t}^{HW}$ defines when and how much hot water is demanded throughout the month’s 2976 time periods. Temperature setpoints $T_{m,t}^{setpoint}$ and ambient conditions $T_{m,t}^{ambient}$ establish the thermal boundary conditions for month $m$. The electricity rate vector $r_{m,t}$ is constructed by mapping peak hours set $H$ to the on-peak rate $r^{on}$ and all other periods to off-peak rate $r^{off}$.

Set Time-Varying Parameters for Month $m$:

Load hot water usage schedule: $U_{m,t}^{HW}$ for all $t \in T$
Load temperature profiles: $T_{m,t}^{setpoint}$, $T_{m,t}^{ambient}$ for all $t \in T$
Set electricity rates: $r_{m,t} = r^{on}$ if $t \in H$, else $r_{m,t} = r^{off}$

Initialize Schedule State for Month $m$:

If $m = 1$: set $S_m^{current} = 1$ (start on default schedule)
Else: $S_m^{current} = S_{m-1}^{current,next}$ (use previous month’s decision outcome)

Set Operational Schedule for Month $m$:

The binary operation permission vector $s_{m,t}$ is derived from the current schedule state $S_m^{current}$. When $S_m^{current} = 1$ (default), the HPWH can operate whenever needed ($s_{m,t} = 1$ for all $t$). When $S_m^{current} = 0$ (TOU-adapted), operation is restricted during peak hours ($s_{m,t} = 0$ when $t \in H$).

\[ s_{m,t} = \begin{cases} 1 & \text{if } S_m^{current} = 1 \text{ (default: always allowed)} \\ 1 & \text{if } S_m^{current} = 0 \text{ and } t \notin H \text{ (TOU: off-peak only)} \\ 0 & \text{if } S_m^{current} = 0 \text{ and } t \in H \text{ (TOU: peak restricted)} \end{cases} \]

Step 2: Run OCHRE Simulation for Month $m$¶

OCHRE executes the building physics simulation for month $m$ using the operational schedule $s_{m,t}$ as a constraint on HPWH operation. For each 15-minute interval $t$ in month $m$, OCHRE determines whether the HPWH can operate based on $s_{m,t}$, then calculates the resulting electricity consumption $E_{m,t}$ and tank temperature $T_{m,t}^{tank}$ considering hot water draws $U_{m,t}^{HW}$, thermal losses, and ambient conditions $T_{m,t}^{ambient}$. The monthly electricity bill is computed by summing the product of consumption and time-varying rates across all time periods in month $m$.

Execute Monthly Simulation for Month $m$:

Input: $U_{m,t}^{HW}$, $s_{m,t}$, $T_{m,t}^{setpoint}$, $T_{m,t}^{ambient}$ for all $t \in T$
Output: $E_{m,t}$, $T_{m,t}^{tank}$ for all $t \in T$

Calculate Monthly Electricity Bill for Month $m$:

\[ C_m^{bill} = \sum_{t \in T} E_{m,t} \cdot r_{m,t} \]

Note that this bill is specific to the HPWH, and does not include other electricity loads in the building. In practice, consumers get a bill that includes all of their electricity usage, and the HPWH bill is a subset of that. However, since other operations remain the same, and we are only changing the HPWH, we don’t need to include other loads in the model.

Step 3: Assess Comfort Performance for Month $m$¶

Comfort assessment for month $m$ begins by identifying time periods where tank temperature $T_{m,t}^{tank}$ falls below the setpoint $T_{m,t}^{setpoint}$ during hot water usage events ($U_{m,t}^{HW} > 0$). For each such period, the thermal energy shortfall $Q_{m,t}^{unmet}$ is calculated as the energy required to heat the delivered water from tank temperature to setpoint temperature, using water density $\rho$ and specific heat $c_p$. This thermal deficit is then converted to electrical energy equivalent $D_{m,t}^{unmet}$ by dividing by the heat pump’s coefficient of performance $COP$ and converting from Joules to kWh. The total comfort penalty for month $m$, $C_m^{comfort}$, monetizes these electrical energy equivalents using the comfort parameter $\alpha$.

Calculate Thermal Unmet Demand for Month $m$:

\[ Q_{m,t}^{\text{unmet}} = \begin{cases} U_{m,t}^{\text{HW}} \cdot \rho \cdot c_p \cdot (T_{m,t}^{\text{setpoint}} - T_{m,t}^{\text{tank}}) & \text{if } T_{m,t}^{\text{tank}} < T_{m,t}^{\text{setpoint}} \text{ and } U_{m,t}^{\text{HW}} > 0 \\ 0 & \text{otherwise} \end{cases} \]

Convert to Electrical Equivalent for Month $m$:

\[ D_{m,t}^{unmet} = \frac{Q_{m,t}^{unmet}}{COP \cdot 3,600,000} \]

Calculate Monthly Comfort Penalty for Month $m$:

\[ C_m^{comfort} = \alpha \cdot \sum_{t \in T} D_{m,t}^{unmet} \]

Step 4: Decision Logic Branch for Month $m$¶

The decision logic branches based on the current schedule state $S_m^{current}$, implementing different information sets and decision criteria for consumers in different situations. See the end of this section for a discussion regarding how consumers might calculate their net savings with the information they have in practice, and how this might differ from the model’s assumptions.

Branch based on current schedule state $S_m^{current}$:

Case A: Currently on Default Schedule ($S_m^{current} = 1$)¶

Consumers on the default schedule in month $m$ evaluate TOU adoption by simulating the alternative schedule and comparing anticipated costs. Since they have no experience with TOU operation, the decision excludes comfort penalties $C_m^{comfort}$, which are unknown at this stage.

Step 4A.1: Simulate Alternative (TOU) Schedule for Month $m$

A temporary TOU schedule is created for month $m$ by setting the operational permissions to restrict peak-hour operation. OCHRE simulates this alternative schedule using the same input conditions ($U_{m,t}^{HW}$, $T_{m,t}^{setpoint}$, $T_{m,t}^{ambient}$) but with the modified operational constraints. This produces the electricity consumption profile $E_{m,t}^{TOU}$ that would result under TOU scheduling in month $m$.

Temporarily set $S_m^{temp} = 0$
Set TOU operational schedule: $s_{m,t}^{temp} = 1$ if $t \notin H$, else $s_{m,t}^{temp} = 0$
Run OCHRE simulation to get $E_{m,t}^{TOU}$
Calculate: $C_m^{bill,TOU} = \sum_{t \in T} E_{m,t}^{TOU} \cdot r_{m,t}$

Step 4A.2: Calculate Anticipated Savings for Month $m$

The anticipated bill savings $\Delta C_m^{anticipated}$ represent the difference between current default schedule costs and projected TOU schedule costs for month $m$. The net anticipated benefit subtracts the one-time switching cost $C^{switch}$ but does not include comfort penalties since the consumer cannot anticipate these impacts.

\[ \Delta C_m^{anticipated} = C_m^{bill} - C_m^{bill,TOU} \]

\[ \text{Net Savings}_m^{anticipated} = \Delta C_m^{anticipated} - C^{switch} \]

Step 4A.3: Make Switching Decision for Month $m$

The binary switching decision $x_m^{switch}$ is determined by whether anticipated net savings are positive in month $m$. If $\text{Net Savings}_m^{anticipated} > 0$, the consumer adopts TOU scheduling ($x_m^{switch} = 1$); otherwise, they remain on the default schedule ($x_m^{switch} = 0$).

\[ x_m^{switch} = \begin{cases} 1 & \text{if } \text{Net Savings}_m^{anticipated} > 0 \\ 0 & \text{otherwise} \end{cases} \]

Case B: Currently on TOU Schedule ($S_m^{current} = 0$)¶

Consumers already on TOU scheduling in month $m$ have experienced both financial and comfort impacts during the current month. Their continuation decision incorporates complete information including realized comfort penalties $C_m^{comfort}$.

Step 4B.1: Simulate Alternative (Default) Schedule for Month $m$

The alternative default schedule simulation determines what electricity costs would have been in month $m$ without TOU restrictions. All operational permissions are set to allow HPWH operation ($s_{m,t}^{temp} = 1$ for all $t$), and OCHRE simulates the resulting consumption $E_{m,t}^{default}$ and costs $C_m^{bill,default}$.

Temporarily set $S_m^{temp} = 1$
Set default operational schedule: $s_{m,t}^{temp} = 1$ for all $t \in T$
Run OCHRE simulation to get $E_{m,t}^{default}$
Calculate: $C_m^{bill,default} = \sum_{t \in T} E_{m,t}^{default} \cdot r_{m,t}$

Step 4B.2: Calculate Realized Performance for Month $m$

The realized savings $\Delta C_m^{realized}$ compare the counterfactual default costs to actual TOU costs experienced in month $m$. The net realized savings subtract both the switching cost $C^{switch}$ and the comfort penalty $C_m^{comfort}$ that was actually experienced during TOU operation.

\[ \Delta C_m^{realized} = C_m^{bill,default} - C_m^{bill} \]

\[ \text{Net Savings}_m^{realized} = \Delta C_m^{realized} - C^{switch} - C_m^{comfort} \]

Step 4B.3: Make Continuation Decision for Month $m$

The continuation decision evaluates whether to remain on TOU scheduling based on complete cost information from month $m$. If realized net savings are non-positive ($\text{Net Savings}_m^{realized} \leq 0$), the consumer switches back to default scheduling ($x_m^{switch} = 1$); otherwise, they continue with TOU ($x_m^{switch} = 0$).

\[ x_m^{switch} = \begin{cases} 1 & \text{if } \text{Net Savings}_m^{realized} \leq 0 \text{ (switch back to default)} \\ 0 & \text{otherwise (stay on TOU)} \end{cases} \]

Step 5: Update State for Next Month¶

The schedule state $S_{m+1}^{current}$ for the next month is determined by the switching decision $x_m^{switch}$ made in month $m$. If switching occurs ($x_m^{switch} = 1$), the state toggles to its opposite value ($1 - S_m^{current}$). If no switching occurs ($x_m^{switch} = 0$), the state remains unchanged. Monthly results for month $m$ including $C_m^{bill}$, $C_m^{comfort}$, $x_m^{switch}$, and $S_m^{current}$ are recorded for annual analysis.

Update Schedule State for Month $m+1$:

\[ S_{m+1}^{current} = \begin{cases} 1 - S_m^{current} & \text{if } x_m^{switch} = 1 \\ S_m^{current} & \text{if } x_m^{switch} = 0 \end{cases} \]

Store Monthly Results for Month $m$:

Record: $C_m^{bill}$, $C_m^{comfort}$, $x_m^{switch}$, $S_m^{current}$
Save for annual analysis and next month’s initialization

Step 6: Monthly Iteration Control¶

The simulation checks whether the annual cycle is complete. If the current month $m < 12$, the month counter increments and the process returns to Step 1 with month $m+1$ and the updated schedule state $S_{m+1}^{current}$. If month 12 is complete, the simulation proceeds to annual evaluation metrics calculation.

Check Simulation Status:

If $m < 12$: increment to month $m+1$, return to Step 1 with $S_{m+1}^{current}$
If $m = 12$: proceed to annual evaluation (Step 7)

Step 7: Annual Evaluation and State Reset¶

For multi-year simulations, the final month’s schedule state $S_{13}^{current}$ becomes the initial state for the following year’s first month, allowing persistence of consumer preferences across years. Before resetting for the next annual cycle, comprehensive evaluation metrics are calculated and key visualizations are generated to assess model performance and consumer behavior patterns.

Step 7.1: Calculate Annual Performance Metrics¶

Financial Performance:

\[ \text{Total Annual Savings} = \sum_{m=1}^{12} \Delta C_m^{realized} \]

\[ \text{Total Switching Costs} = \sum_{m=1}^{12} x_m^{switch} \cdot C^{switch} \]

\[ \text{Total Comfort Penalty} = \sum_{m=1}^{12} C_m^{comfort} \]

\[ \text{Net Annual Benefit} = \text{Total Annual Savings} - \text{Total Switching Costs} - \text{Total Comfort Penalty} \]

Behavioral Metrics:

\[ \text{TOU Adoption Rate} = \frac{\sum_{m=1}^{12} (1 - S_m^{current})}{12} \times 100\% \]

\[ \text{Annual Switches} = \sum_{m=1}^{12} x_m^{switch} \]

\[ \text{Average Monthly Bill} = \frac{1}{12} \sum_{m=1}^{12} C_m^{bill} \]

System Performance:

\[ \text{Peak Load Reduction} = \frac{\sum_{m=1}^{12} \sum_{t \in H} (E_{m,t}^{baseline} - E_{m,t})}{\sum_{m=1}^{12} \sum_{t \in H} E_{m,t}^{baseline}} \times 100\% \]

Step 7.2: Generate Key Visualizations¶

A. Annual Decision Timeline

Line plot showing $S_m^{current}$ across months with switching events $x_m^{switch}$ marked
Purpose: Visualize adoption patterns and decision stability

B. Monthly Cost Decomposition

Stacked bar chart with $C_m^{bill}$, $C_m^{comfort}$, and switching costs for each month
Purpose: Show relative impact of each cost component

C. Performance Scatter Plot

X-axis: $C_m^{comfort}$, Y-axis: $\Delta C_m^{realized}$, color-coded by $S_m^{current}$
Purpose: Identify trade-offs between savings and comfort

D. Load Profile Heatmap

2D plot: months (x-axis) vs. hours (y-axis), color intensity = average $E_{m,t}$
Purpose: Visualize seasonal and daily load shifting patterns

Step 7.3: Reset for Next Year¶

Prepare for Next Year:

Set $S_1^{current} = S_{13}^{current}$ (carry forward final state)
Clear monthly arrays: $\{C_m^{bill}, C_m^{comfort}, x_m^{switch}, S_m^{current}\}_{m=1}^{12}$
Update annual parameters (e.g., rate changes, equipment degradation)
Export annual metrics to results database
Return to Step 1 for new annual cycle with $m = 1$

This evaluation framework provides both quantitative metrics for model validation and intuitive visualizations for understanding consumer behavior patterns and system-wide impacts.

State space diagram¶

stateDiagram-v2
  [*] --> S_default: First month
  S_default --> S_TOU: Switch to TOU\nNet Savings^{anticipated} > 0
  S_default --> S_default: Stay\nNet Savings^{anticipated} ≤ 0
  S_TOU --> S_default: Switch back to Default\nNet Savings^{realized} ≤ 0
  S_TOU --> S_TOU: Stay\nNet Savings^{realized} > 0
  S_default: Default Schedule (S^{current}=1)
  S_TOU: TOU-adapted Schedule (S^{current}=0)

Consumer Information and Decision-Making Reality¶

How Consumers Actually Estimate HPWH Schedule Switching Benefits in Practice¶

In the real world, consumers who are already on TOU rates cannot run sophisticated building simulations to predict whether changing their water heater schedule will save money. Instead, they rely on simplified mental models, basic calculations, and trial-and-error to estimate potential benefits from shifting their HPWH operation to off-peak hours.

The Typical Consumer Journey¶

Monthly Bill Review Phase:

Sarah receives her TOU electricity bill showing $92 this month, with a breakdown: $38 from peak hours (2-8 PM) and $54 from off-peak. She notices her electric water heater is one of her largest energy users. She wonders: “What if I could get my water heater to run mostly during off-peak times?”

Basic Calculation Attempt:

Sarah looks at her bill and sees she’s paying $0.28/kWh during peak vs. $0.12/kWh off-peak. She estimates her water heater uses about 800 kWh/month and reasons: “If half of that is currently happening during peak hours, that’s 400 kWh × ($0.28 - $0.12) = $64 potential savings per month. But realistically, maybe I can shift 70% of peak usage to off-peak, so perhaps $45 savings?”

Social Information Gathering:

At work, Sarah asks her colleague Tom about his programmable water heater. Tom says: “I set mine to heat from 10 PM to 6 AM. My bill went down maybe $20-25 per month, though sometimes the water isn’t quite as hot for evening dishes. It’s a trade-off, but worth it for the savings.”

Switching Cost Assessment:

Sarah considers the total effort required: analyzing her current usage patterns (2 hours), figuring out the water heater programming (2-3 hours), and periodically checking bills to ensure it’s working. She estimates this represents about $35 worth of her time and mental energy. With potential monthly savings of $20-30, the payback period is reasonable: “Even if I only save $20/month, that’s $240/year for maybe 5 hours of total effort.”

What Consumers Actually Calculate¶

Peak Usage Estimation:

Look at TOU bill breakdown or estimate major appliance usage during peak hours
Apply simple fractions: “Maybe 40% of my water heating happens during peak time”

Rate Differential Application:

Calculate potential savings as: (estimated peak kWh) × (peak rate - off-peak rate) × (shifting efficiency)
Use round numbers: “If I shift 300 kWh from $0.28 to $0.12, that’s $48 savings”

Potential Switching Cost Components:

Information cost: Time analyzing bills and researching strategies
Implementation cost: Programming water heater controls and trial-and-error
Monitoring cost: Ongoing bill checking and adjustments

Implications for Model Design¶

This realistic decision-making process suggests consumers estimate $\Delta C^{anticipated}$ using:

\[ \hat{\Delta C}_m^{anticipated} = \hat{E}_m^{peak,WH} \times (r^{on} - r^{off}) \times \phi \]

Where:

$\hat{E}_m^{peak,WH}$ = consumer’s guess of peak-hour water heating usage
$\phi$ = expected shifting effectiveness (0.6-0.8)

And evaluate switching against realistic transaction costs:

\[ C^{switch} = \$35 \text{ (representing 3-5 hours of consumer effort)} \]

The model should reflect that consumers make switching decisions based on rough mental calculations and significant uncertainty about both benefits and comfort impacts, not sophisticated building physics simulations.

5. References¶

This section lists references and resources for further information and context regarding the model and its implementation.