*3.1. Response to Dynamic Pricing*

The household appliances can be broadly categorized into four groups, i.e., non-controllable, buffer appliances, time shifting and curtailable appliances [32]. The non-controllable appliances are mostly commonly known as the base load and include the devices that do not offer controllability for different levels of price. Buffer appliances represent devices with integrated storage facilities in terms of thermal (e.g., freezers, heat pump) or electro-chemical energy (e.g., batteries, EV). Time-shifting appliances can shift their consumption in time based on price and include the washing machine, dishwasher, etc. Finally, the curtailable appliances refer to the devices that can be interrupted based on predefined contracts for different purposes (e.g., solar PV system for voltage violation or network congestion). Each type of these appliances would react differently to the incoming price levels. The HA tries to optimize the energy cost of the house by utilizing the available flexible loads, and the associated optimization problem could be expressed as:

$$\min \sum\_{a=1}^{|A|} \sum\_{t=1}^{N\_T} p\_t \times P\_{t,a} \times \Delta t \tag{1}$$

where *A* is the set of all appliances within a single household, *NT* is the number of time steps in a day, *pt* is the price at time *t, Pt,a* is the power of appliance *a* at time *t* and Δ*t* is the duration of each time step.

The buffer and time shifting appliances need to optimize their profiles based on the dynamic price. Since the devices are categorized in different groups, their energy use can be considered independent of each other. Equation (1) can thus be re-written as:

$$\sum\_{a=1}^{|A|} \min \sum\_{t=1}^{N\_T} p\_t \times P\_{t,a} \times \Delta t. \tag{2}$$

The behavior of the buffer appliances can be formulated as:

$$\min \sum\_{t}^{N\_T} p\_t \times P\_{t,a} \times \Delta t \; \forall a \in A^{bf} \tag{3}$$

subject to,

$$\sum\_{k+\mathbf{x}\_a}^{t=k} P\_{t,a} = \sum\_{k+\mathbf{x}\_a}^{t=k} P\_{t,a}^0 \,\,\forall a \in A^{bf}, k \in T \tag{4}$$

$$\mathbf{m} \mathbf{m} \mathbf{n}\_T P^0\_{t,a} < P\_{t,a} < \mathbf{m} \mathbf{x}\_T P^0\_{t,a} \; \forall a \in A^{bf}, t \in T \tag{5}$$

where *xa* is the maximum buffer time for appliance *a*, *P*<sup>0</sup> *<sup>t</sup>*,*<sup>a</sup>* is the original load of appliance *a* at time *t*, *T* is the set of the time steps in a day and *Abf* is the set of the buffer appliances. The constraints in Equations (4) and (5) control that the total energy consumed by the device will remain constant and buffer time, *xa*, could be altered depending on the price.

For time shifting devices, the behavior can be represented as the following optimization problem:

$$\min \sum\_{t=1}^{N\_T} p\_t \times P\_{t+\tau,\mu} \times \Delta t \quad \forall a \in A^{ts} \tag{6}$$

subject to,

$$\sum\_{t=1}^{N\_T} P\_{t,a} = \sum\_{t=1}^{N\_T} P\_{t+\tau,a} \quad \forall a \in A^{ts} \tag{7}$$

$$
\tau\_{\min,a} < \pi < \tau\_{\max,a} \quad \forall a \in A^{ts} \tag{8}
$$

where τ is the time shift of the appliance, τ*min*,*<sup>a</sup>* and τ*max*,*<sup>a</sup>* are the limits of the maximum allowable time shift and *Ats* is the set of all time shifting appliances. The optimization problem of the time shifting appliances needs to use the time shift τ as the decision variable rather than the power at each time instant.

In this work, PV systems have been considered as curtailable only, and their behavior has been considered inelastic to dynamic price signals. More information of the device level modeling can be found in [6].
