The optimal scheduling problem is modeled as a mixed-integer linear programming (MILP) problem, where the variables to be determined are the status (on=1 or off=0) of each device in each time step with respect to the different energy carriers. In particular, *δ*_{a,s,t,l} is a binary variable indicating the operation status (0 or 1) of appliance *a* requiring level *l* from the energy carrier *s* at time step *t*. We simulate a 24 h horizon with 15-min time steps. Five days with different thermal load are simulated with four possible configurations.

We assume that CO_{2} signals with quarterly resolution are known at the beginning of the simulated day, so that the supply and demand can be optimally scheduled for the whole day. This may actually not be the case in a real scenario, where data about the German generation mix are available with a couple of hours of delay. However, it is possible to forecast such data with accuracy of around 10%, for example by using those of the previous 24 h as suggested in (Kristinsdóttir et al. 2013). Moreover, according to the proposed problem formulation, devices may not be scheduled to the next day and thus may not benefit from low emissions after the evening peak. To contain the effects of such limitation, main shiftable loads (i.e., DW, WM, and TD) are assumed to have a 24 h operation window and thus they may be scheduled during night period.

### Objective function

The daily *C**O*_{2} emissions to be minimized are defined as follows:

$$ min \left[ \Delta \text{t}\cdot \sum\limits_{t=1}^{96}\left({EF}_{\text{e},t}\sum\limits_{a\in A}^{} \frac{P_{a,\text{e,}t}}{\eta_{\text{e},a}}+{EF}_{\text{h}}\sum\limits_{a\in A}^{} \frac{P_{a,\text{h,}t}}{\eta_{\text{h},a}} + {EF}_{\text{e},t}\sum\limits_{a\in A}^{} \frac{P_{a,\text{eh,}t}}{\eta_{\text{e},a}} \right) \right] $$

(2)

where *P*_{a,e,t} is the power consumption of appliance *a* at time step *t* when operating in electricity-only mode, while *P*_{a,h,t}, and *P*_{a,eh,t} are the heat and electricity demands when operating in hybrid mode. *Δ*t is the duration of one time step, i.e., a quarter of an hour.With respect to the kettle, we consider a 2200 W-device that operates for 5 min out of one time step. By doing so, the energy consumption is realistically set at 183 Wh (Wood and Newborough 2003) and the emissions are calculated accordingly.

The power and/or heat demands at each time step *t* for each appliance *a* are defined as follows:

$$ P_{a,s,t} = \sum\limits_{l=1}^{\gamma_{a}}\delta_{a,s,t,l}\cdot P_{a,s,l} \hspace{5mm} \forall a \in A, \forall s \in S, \forall t \in T $$

(3)

where *P*_{a,s,l} is the demand of source *s* equal to level *l* for appliance *a*, *T* is the set of time steps, and *S* is the set of energy carriers (electricity-only *e*, heat *h*, electricity in hybrid mode *eh*). Given the demand profile of a device during its operation, the required power or heat in each quarter of hour corresponds to a level *l*. We define *A* as the set of all appliances, which does not include the thermal load indicated as *SW*.

The objective function is subjected to several constraints as described next.

### Constraints

Hybrid appliances can work in electricity-only or hybrid mode, that means, they require electricity and/or hot water/gas in a time step *t* (Equation 4). Additionally, when working in hybrid mode, both electricity and heat have to be (un)used at the same time (Equation 5)

$$ \sum_{s \in S} \sum_{l=1}^{\gamma_{a}} \delta_{a,s,t,l} \le 2 \hspace{5mm} \forall a \in A, \forall t \in T $$

(4)

$$ \delta_{a,\mathrm{h},t,l} = \delta_{a,\text{eh},t,l} \hspace{5mm} \forall a \in A, \forall t \in T, \forall l \in [1,\gamma_{a}] $$

(5)

The thermal load can work in electricity-only or gas-only modes, hence in each time step *t* it can use one energy carrier at the most (Equation 6) and no hybrid operation is possible (Equation 7). Moreover, one single level of demand can be satisfied at each time step.

$$ \sum_{s \in S} \sum_{l=1}^{\gamma_{\text{SW}}} \delta_{{\text{SW}},s,t,l} \le 1 \hspace{5mm} \forall t \in T $$

(6)

$$ \delta_{{\text{SW}},\text{eh},t,l}=0 \hspace{5mm} \forall t \in T, \forall l \in [1,\gamma_{\text{SW}}] $$

(7)

The consumer sets an operation window for each appliance [ *α*_{a}, *β*_{a}], and no operation is allowed outside this interval (Equation 8). Additionally, all appliances have to complete their operation of length *γ*_{a} within the operation window and they are assumed to run in non-interruptible mode. Moreover, the levels of demand have to be satisfied consecutively. These constraints are summarized in Equation 9.

$$ \begin{aligned} \delta_{a,s,t,l} = 0 \hspace{5mm} \forall a \in A \cup SW, \forall s\in S, \forall l \in [1,\gamma_{a}], \forall t \in T \setminus [\alpha_{a},\beta_{a}] \end{aligned} $$

(8)

$$ \begin{aligned} \delta_{a,s,t,l} = \delta_{a,s,t-1,l-1} \hspace{5mm} \forall a \in A \cup SW, \forall s\in S, \forall l \in [1,\gamma_{a}], \forall t \in [\alpha_{a},\beta_{a}] \end{aligned} $$

(9)

The total power imported from the grid is limited to *P*_{e,MAX}=8 kW in electricity-only mode and to 3 kW in the hybrid one.

$$ \sum\limits_{a \in A \cup SW}^{} \sum\limits_{s \in S\setminus h}^{} P_{a,s,t} \le P_{\mathrm{e,MAX}} \hspace{5mm} \forall t \in T $$

(10)

Similarly, the output of the gas-burning boiler is limited to *P*_{b,MAX}=15 kW. We indicate with *A*_{hw} the set of appliances using hot water in hybrid mode, namely the washing machine, the dryer, and the dishwasher.

$$ \sum\limits_{a \in A_{\text{hw}} \cup SW}^{} P_{a,\mathrm{h},t} \le P_{\mathrm{b,MAX}} \hspace{5mm} \forall t \in T $$

(11)

Moreover, we assume that the dryer has to run after the washing machine ends, hence:

$$ \sum\limits_{\tau = 1}^{t-1} \sum\limits_{l=1}^{\gamma_{\text{WM}}} \sum\limits_{s \in S\setminus eh} \delta_{{\text{WM}},s,\tau,l} \ge \gamma_{\text{WM}} \cdot \sum\limits_{s \in S\setminus eh} \delta_{{\text{TD}},s,t,0} \hspace{5mm} \forall t \in T $$

(12)