4.2.2. Constraints

#### 4.2.2.1. Lithium-Ion Degradation Model

The battery degradation model is valid for both the EV and the BES, and therefore, the identifier *X* = *EV* = *BES* is used. In order to calculate the capacity lost per time step (Δ*Etot X* (*t*)), the battery degradation model presented in [44] is used. The model is semi-empirical based on 18650 Nickel-Manganese-Cobalt (NMC) cells. It takes into account temperature, current rate (*Icell X* (*t*)), and ampere-hours processed (*Icell X* (*t*)*δt*). In this study, the cell temperatures are assumed constant at 35 ◦C, assuming that the EV has a battery temperature control system and because the inside ambient temperature setpoint of the heat pump varies between 18◦ and 20.5 ◦C [45]. A distinction between cyclic and calendar ageing is made, denoted as Δ*Ecycle X* (*t*) and Δ*Eca<sup>l</sup> X* (*t*). Since calendar ageing is mostly dependent on time and temperature [44] and cell temperature is assumed constant, the equation can be simplified to a constant degradation per time step. Here, a minimum lifetime of 5 years is assumed. This is shown in Equation (12).Furthermore, the model is based on the behaviour of a single cell. Therefore the EV/BES power needs to be scaled into the voltage and current of a single cell. To do this, the open-circuit voltage of an NMC cell is used, which can be described by the curve fitted equation shown in Equation (9) [46]. Here, *SoCx*(*t*) denotes the state of charge of *x* at time *t*. The resulting curve is shown in Figure 4. Furthermore, it is assumed that *Nparallel X* by *Nseries X* of these cells are placed in parallel and series respectively, such that the total open-circuit voltage *Voc*,*<sup>X</sup>* resembles existing EV/BES systems [33,34].

$$V\_{\text{oc},X}(t) = N\_X^{\text{sgrics}} \left( a\_1 e^{b\_1 \text{Soc}\_x(t)} + a\_2 e^{b\_2 \text{Soc}\_x(t)} + a\_3 \text{Soc}\_x(t)^2 \right) \tag{9}$$

$$\dot{q}\_X^{\text{cell}}(t) = \frac{P\_X(t)}{N\_X^{\text{parallel}} V\_{\text{oc},X}(t)} \qquad , \forall \ t \tag{10}$$

Then, using the calculated cell voltage and current from Equation (9) and (10), the lost capacity per cell can be calculated according to Equations (11) and (13) [44]. Note that the model presented in [44] calculates the percentage of lost charge in ampere hour and is therefore multiplied with *Emax BES* 100 in order to ge<sup>t</sup> the actually lost capacity. The values of the curve fitted parameters in Equations (9)–(13) can be found in Table 2. 

$$
\Delta E\_X^{cycle}(t) = \left( c\_1 e^{c\_2|i\_X^{cell}(t)|} |i\_X^{cell}(t)| \Delta t \right) \frac{E\_{BES}^{max}}{100} \qquad , \forall \ t \tag{11}
$$

$$
\Delta E\_X^{\rm cal}(t) = \left( c\_3 \sqrt{t} e^{-24k/RT} \right) \frac{E\_{BES}^{\rm max}}{100} = \left( c\_4 \Delta t \right) \frac{E\_{BES}^{\rm max}}{100} \qquad , \forall \ t \tag{12}
$$

$$
\Delta E\_X^{tot} = \sum\_{t=0}^T \left( \Delta E\_X^{cycle}(t) + \Delta E\_X^{cal}(t) \right) \tag{13}
$$

**Figure 4.** Open circuit voltage for a Nickel–Manganese–Cobalt (NMC) battery cell [46]: Since both the electric vehicle and the stationary battery are made of NMC technology, this cell voltage is applicable for both.

**Table 2.** Battery model parameters.


#### 4.2.2.2. Battery Energy Storage Constraints

The BES power *PBES*(*t*) is limited between [−10, 10] kW. Additionally, the maximum power *Pmax BES* (*t*) is SoC dependent such that the maximum power drops linearly below a SoC of 10% (*Ddis* = 0.1) and above a SoC of 80% (*Dch* = 0.8), representing the constant-current constant-voltage regions of a battery. This is ensured using Equations (14)–(20). The round-trip efficiency is considered constant over power and lifetime and is equal to 95 % [34]. Therefore, the efficiency of a single charge/discharge cycle equals *ηch*/*dis* = √0.95 = 0.975. Due to this efficiency, the model recognizes the loss of energy and therefore prevents both *Pneg BES*(*t*) and *Ppos BES*(*t*) from having nonzero values at the same time.

$$P\_{BES}^{\rm pos}(t) \le P\_{BES}^{\rm max}(t) \qquad , \forall \ t \tag{14}$$

$$P\_{BES}^{\max}(t) \le P\_{BES}^{\text{rated}}(t) \qquad , \forall \ t \tag{15}$$

$$P\_{BES}^{\max}(t) \le \frac{P\_{BES}^{\max}}{1 - D\_{ch}} \left(\frac{E\_{BES}(t)}{E\_{BES}^{\max}} - 1\right) \qquad , \forall \ t \tag{16}$$

$$P\_{BES}^{m\text{cyf}}(t) \le P\_{BES}^{m\text{in}}(t) \qquad , \forall \ t \tag{17}$$

$$P\_{BES}^{\text{min}}(t) \le P\_{BES}^{\text{rated}} \qquad , \forall \ t \tag{18}$$

$$P\_{BES}^{min}(t) \le \frac{P\_{BES}^{rated}}{D\_{dis}} \frac{E\_{BES}(t)}{E\_{BES}^{max}} \qquad , \forall \ t \tag{19}$$

$$P\_{BES}(t) = \eta\_{ch} P\_{BES}^{\text{pos}}(t) - \frac{1}{\eta\_{dis}} P\_{BES}^{\text{neg}}(t) \qquad , \forall \ t \tag{20}$$

In the above constraints as well as all the following constraints, the superscripts "max", "min", "pos", "neg", and "rated" declare the maximum allowable, minimum allowable, actual positive, actual negative, and rated powers.The energy stored inside the BES *EBES*(*t*) can then be calculated according to Equation (21). Here, the BES capacity is fixed for *t* = 1 and *t* = *tfinal* at capacities *Einit BES* = *Efinal BES* , respectively, in order to have a fair comparison between costs.

$$E\_{BES}(t) = \begin{cases} E\_{BES}^{init} & \text{for } t = 1\\ E\_{BES}(t-1) + P\_{BES}(t)\Delta t, & \text{for } 1 < t < t\_{final} \\ E\_{BES}^{final} & \text{for } t = t\_{final} \end{cases} \tag{21}$$

Furthermore, the SoC of the BES is calculated using Equation (22). Here, *Elimit BES* (*t*) is a variable which represents the actual maximum capacity at time t, which equals the initial maximum capacity *Emax BES* minus the capacity lost by cycling for that time step <sup>Δ</sup>*EBES*(*t*). This is modelled using Equation (23) and Equation (24).

$$SoC\_{BES}(t) = \frac{E\_{BES}(t)}{E\_{BES}^{limit}(t)} \qquad , \forall \ t \tag{22}$$

$$E\_{BES}^{limit}(t) = \begin{cases} E\_{BES}^{max} & \text{for } t = 1\\ E\_{BES}^{limit}(t-1) - \Delta E\_{BES}(t), & \text{for } t > 1 \end{cases} \tag{23}$$

$$E\_{BES}(t) \le E\_{BES}^{limit}(t) \qquad , \forall \ t \tag{24}$$

#### 4.2.2.3. Electric Vehicle Constraints

The electric vehicle constraints are given in Equations (25)–(36), where the constraints up to Equation (34) are similar to the BES constraints. The electric vehicle is assumed to be unavailable from *tdepart* = 08:00 till *tarrive* = 18:00 during the day; this is denoted by the binary parameter *EVav*(*t*), which equals 0 for *tdepart* ≤ *t* ≤ *tarrive*. These departure and arrival times are based on the distribution, as presented in [47]. During that time, it is estimated that the EV is commuting between work and home, where a single trip is 30 km with an efficiency of 15 kWh/100 km [48], resulting in a 9 kWh decrease in charge at arrival; note that the arrival time and charge are only estimations and that possible errors will be compensated after arrival as a result of the moving horizon window.

$$P\_{EV}^{\rm pvs}(t) \le P\_{EV}^{\rm max}(t) \qquad , \forall \ t \tag{25}$$

$$P\_{EV}^{\text{max}}(t) \le P\_{EV}^{\text{rated}}(t) \qquad , \forall \ t \tag{26}$$

$$P\_{EV}^{max}(t) \le \frac{P\_{EV}^{max}}{1 - D\_{ch}} \left(\frac{E\_{EV}(t)}{E\_{EV}^{max}} - 1\right) \qquad , \forall \; t \tag{27}$$

$$P\_{EV}^{\rm{M\%}}(t) \le P\_{EV}^{\rm{min}}(t) \qquad , \forall \ t \tag{28}$$

$$P\_{EV}^{min}(t) \le P\_{EV}^{rated} \qquad , \forall \ t \tag{29}$$

$$P\_{EV}^{min}(t) \le \frac{P\_{EV}^{rated}}{D\_{dis}} \frac{E\_{EV}(t)}{E\_{EV}^{max}} \qquad , \forall \ t \tag{30}$$

$$P\_{EV}(t) = EV\_{av}(t) \left( \eta\_{ch} P\_{EV}^{pos}(t) - \frac{1}{\eta\_{dis}} P\_{EV}^{ncg}(t) \right) \qquad , \forall \ t \tag{31}$$

$$So\mathbb{C}\_{EV}(t) = \frac{E\_{EV}(t)}{E\_{EV}^{limit}(t)}\qquad , \forall \ t\tag{32}$$

$$E\_{EV}^{limit}(t) = \begin{cases} E\_{EV}^{max}, \text{for } t = 1\\ E\_{EV}^{limit}(t-1) - \Delta E\_{EV}(t), \text{for } t > 1 \end{cases} \tag{33}$$

$$E\_{EV}(t) \le E\_{EV}^{limit}(t) \qquad , \forall \ t \tag{34}$$

Finally, the user can state a minimum departure charge *Edepart* and departure time *tdepart*, which ensures that the EV always has enough charge at the time of departure. This is modeled using Equation (36):

$$E\_{EV}(t) = \begin{cases} E\_{EV}^{init} & \text{for } t = 1 \\ E\_{EV}(t-1) + P\_{EV}(t)\Delta t, & \text{for } t \le t\_{depart} \\\\ & \& \text{ t > t\_{arrive}} \end{cases} \tag{35}$$

$$E\_{EV}(t) \ge E\_{EV}^{depart}, \text{for } t = t\_{depart} \tag{36}$$

#### 4.2.2.4. Power Balance Constraints

For the given system (Figure 1), two power balances exist: 1. on the DC link of the multi-port converter. Here, a positive inverter power *Pinv*(*t*) equals feeding power to the grid. 2. The second power balance exists on the AC side between the inverter and the meter. This is modeled using Equations (37) and (38).

$$P\_{\rm inv}(t) = P\_{\rm PV}(t) - P\_{\rm BES}(t) - P\_{\rm EV}(t) \qquad , \forall \ t \tag{37}$$

$$P\_{grid}(t) = \eta\_{inv} P\_{inv}(t) - P\_{load}(t) \qquad , \forall \ t \tag{38}$$

Here, the total load *Pload*(*t*) consists of the load from all appliances in the building *Pappl*(*t*) and the power required for heating *Pheat*(*t*), in the form of a heat pump. In this study, both *Pappl*(*t*) and *Pheat*(*t*) are considered non-flexible.

$$P\_{load}(t) = P\_{appl}(t) + P\_{heat}(t) \qquad , \forall \ t \tag{39}$$

### 4.2.2.5. Grid Constraints

The resulting grid power *Pgrid*(*t*) differentiates between positive and negative grid powers since both have different prices. This is done using the efficiency *ηcable*, which models the power loss in the cable between the meter and converter. This efficiency is assumed to equal 99%. However, more importantly, it ensures that the *Pbuy grid*(*t*) and *Psell grid*(*t*) do not have nonzero values simultaneously, as the efficiency loss is recognized. This allows for grid power arbitration without the use of binary variables, drastically increasing the solving time. Here, *Pmax grid* is the maximum power of a 3-phase 25A connection.

$$P\_{grid}^{scl}(t) \le P\_{grid}^{max} \qquad , \forall \ t \tag{40}$$

$$P\_{grid}^{hyy}(t) \le P\_{grid}^{max} \qquad , \forall \ t \tag{41}$$

$$P\_{grid}(t) = \eta\_{cable} P\_{grid}^{sell}(t) - \frac{1}{\eta\_{cable}} P\_{grid}^{huy}(t) \qquad , \forall \ t \tag{42}$$

#### 4.2.2.6. Regulation Market Constraints

Part of the objective function is the revenue obtained from reserving the capacity for primary frequency regulation. Here, it is assumed that the system is part of a smart grid as described in Section 2. Using Equations (43)–(47), it is ensured that the maximum available capacity for regulation does not exceed the actual maximum capacity in the system. Here, it is important to note that the maximum available EV/BES capacity is SoC dependent. Therefore, the available up/down capacity per converter port *Pxup*/*dwn*(*t*) (*x* = EV/BES/PV) is denoted using *Pmax*/*min x* (*t*), as calculated in Sections 4.2.2.2 and 4.2.2.3. Similarly, it has been done for down regulation in Equations (48)–(53).

$$P\_{up}^{EV}(t) \le EV\_{av}(t) \left( P\_{EV}^{min}(t) + P\_{EV}(t) \right) \qquad , \forall \ t \tag{43}$$

$$P\_{up}^{BES}(t) \le P\_{BES}^{min}(t) + P\_{BES}(t) \qquad , \forall \ t \tag{44}$$

$$P\_{\rm up}(t) \le \eta\_{\rm inv} \left( P\_{\rm up}^{EV}(t) + P\_{\rm up}^{BES}(t) \right) \qquad , \forall \, t \tag{45}$$

$$P\_{up}(t) \le P\_{inv}^{rated}(t) \qquad , \forall \ t \tag{46}$$

$$P\_{up}(t) \le P\_{grid}^{\max}(t) + P\_{load}(t) \qquad , \forall \ t \tag{47}$$

$$P\_{dwn}^{EV}(t) \le EV\_{av}(t) \left( P\_{EV}^{max}(t) - P\_{EV}(t) \right) \qquad , \forall \; t \tag{48}$$

$$P\_{dwu}^{BES}(t) \le P\_{BES}^{min}(t) - P\_{BES}(t) \qquad , \forall \ t \tag{49}$$

$$P\_{down}^{PV}(t) \le P\_{PV}(t) \qquad , \forall \ t \tag{50}$$

$$P\_{duvn}(t) \le \eta\_{inv} \left( P\_{EV}^{duvn}(t) + P\_{BES}^{duvn}(t) + P\_{PV}^{duvn}(t) \right) \qquad , \forall \ t \tag{51}$$

$$P\_{duvn}(t) \le P\_{inv}^{rated}(t) \qquad , \forall \ t \tag{52}$$

$$P\_{dwn}(t) \le P\_{glrid}^{max}(t) \qquad , \forall \ t \tag{53}$$

In case of symmetric reserve offers, Equation (54) should also be used.

$$P\_{\rm up}(t) = P\_{\rm down}(t) \qquad , \forall \; t \tag{54}$$

### 4.2.2.7. Inverter Constraints

To account for the efficiency of the inverter, the inverter power is also split into positive and negative parts. The inverter efficiency is assumed equal for both directions of power: 97% and over the entire power range [24].

$$P\_{inv}^{m\text{cyf}}(t) \le P\_{inv}^{max} \qquad , \forall \ t \tag{55}$$

$$P\_{inv}^{pos}(t) \le P\_{inv}^{max} \qquad , \forall \ t \tag{56}$$

$$P\_{inv}(t) = \eta\_{inv} P\_{inv}^{\text{pos}}(t) - \frac{1}{\eta\_{inv}} P\_{inv}^{\text{neg}}(t) \qquad , \forall \ t \tag{57}$$

### 4.2.2.8 Photovoltaic Constraints

As part of demand-side managemen<sup>t</sup> as well as for cases with a negative feed-in tariff, it should be possible to curtail PV power. Therefore, in order to allow PV power curtailment, Equation (58) is introduced. Here, the efficiency of the maximum power point tracker is assumed to be *ηmpp<sup>t</sup>* = 98% [49]. Here, *Pf orecast PV* (*t*) denotes the forecasted PV power. This concludes the optimization model section. In the next section, the proposed real-time control is discussed.

$$P\_{PV}(\mathbf{t}) \le \eta\_{mppt} P\_{PV}^{forcast}(\mathbf{t}) \qquad , \forall \ \mathbf{t} \tag{58}$$

#### *4.3. Moving Horizon Window and Real-Time Control*

In Section 4.2, the optimal charging algorithm is discussed. The goal of the optimal charging algorithm is to find the optimal charging schedule within a 24-h optimization window while anticipating future demand and supply based on forecasts and taking into account battery degradation, and primary frequency regulation reserve. However, as shown in Figure 2, the resulting forecasting errors can still be in the range of several kWs. To minimize the effect of these errors, a moving horizon model predictive controller is implemented, which reoptimizes every 15 min (i.e., one optimization timestep). Additionally, a new forecast is obtained every hour to improve accuracy. To compensate for the errors within these 15-min timesteps, such as PV power, SoC estimation, and EV time of arrival, a real-time control scheme is implemented. Here, a rule-based control scheme is implemented such that it can act almost instantly. It should be noted that the real-time control only deals with the errors on top of what is optimally scheduled. For example, if the battery is directly charging from PV power in the optimal solution but the PV power turns out to be less than forecasted, the battery power can be changed accordingly. If no real-time control is implemented, this error needs to be compensated using grid power to maintain the power balance on the DC-link inside the multi-port converter, which could lead to nonoptimal solutions. Figure 5 shows the flowchart of the moving horizon control and the real-time control scheme. First, the irradiance forecasts are obtained, and the resulting PV power is calculated. Then, after the optimization, the output at *t* = 1 is saved and the sample rate is increased 15 times, such that the resolution is now one minute. The interpolated variables are indicated with index *k*. Furthermore, superscripts *f c* and *act* denote the forecasted and actual powers, respectively. In practice, the resolution can be as small as computation time allows to ge<sup>t</sup> actual real-time operation. The new real-time charging schedule is determined based on the amplitude of the error, the available EV/BES power, and the current electricity price. The error is calculated according to Equations (59)–(61).

$$
\Delta P\_{PV}(k) = P\_{PV}^{fc}(k) - P\_{PV}^{act}(k) \tag{59}
$$

$$
\Delta P\_{load}(k) = P\_{load}^{fc}(k) - P\_{load}^{act}(k) \tag{60}
$$

$$P\_{error}(k) = \Delta P\_{load}(k) - \Delta P\_{PV}(k)\tag{61}$$

Here, a positive error means an excess of power. For example, when the actual *PV* power is higher as forecasted (i.e., Δ *PPV*(*k*) > 0) while the actual load is lower as anticipated (i.e., Δ *Pload*(*k*) < 0), the resulting error is positive *Perror*(*k*) > 0. Next, the power limitations of the EV/BES converter at time k are determined based on power rating, SoC, and power balance inside the multi-port system. This is done using Equations (62)–(65). A rule-based control scheme then determines how the error should be compensated based on *Perror*, the power limitations, current electricity price, and the mean electricity price calculated according to Equation (66), as shown in Figure 5. The goal of the real-time control scheme is to prevent feeding in power to the grid at times of low electricity prices or drawing power during high electricity prices. For example, when the error is positive (meaning an excess of power) and the electricity price *<sup>λ</sup>buy* is above average ( *λmean*), the power is fed to the grid. If the price would be below average, the control first checks whether the BES can absorb the power and, if not, whether the EV can absorb the power; if both are not able to absorb the excess power, it is fed to the grid. After the real-time powers have been calculated, the actual degradation is calculated and the corresponding variables are adjusted. This will then be used for initializing the next optimization instance.

$$P\_{BES}^{\max, \text{new}}(k) = \max\left(0, \min\left(P\_{BES}^{\max}, \frac{E\_{BES}(k) - E\_{BES}^{\min}}{\Delta k}, P\_{inv}^{\max} - P\_{PV}(k) + P\_{EV}(k)\right)\right) \tag{62}$$

$$P\_{BES}^{\max, \text{pos}}(k) = \max\left(0, \min\left(P\_{BES}^{\min}, \frac{E\_{BES}^{\max} - E\_{BES}(k)}{\Delta k}, P\_{\text{inv}}^{\max} + P\_{PV}(k) - P\_{EV}(k)\right)\right) \tag{63}$$

$$P\_{EV}^{\max, \text{ncg}}(k) = \max\left(0, \min\left(P\_{EV}^{\max}, \frac{E\_{EV}(k) - E\_{EV}^{\min}}{\Delta k}, P\_{inv}^{\max} - P\_{PV}(k) + P\_{RES}(k)\right)\right) \tag{64}$$

$$P\_{EV}^{\max, \text{pos}}(k) = \max\left(0, \min\left(P\_{EV}^{\min}, \frac{E\_{EV}^{\max} - E\_{EV}(k)}{\Delta k}, P\_{inv}^{\max} + P\_{PV}(k) - P\_{RES}(k)\right)\right) \tag{65}$$

$$
\lambda\_{\text{mcam}} = \frac{\sum\_{t=1}^{t=T} \lambda\_{\text{buy}}(t)}{T}, \qquad T \in [t, t+24h] \tag{66}
$$

**Figure 5.** Moving horizon and real-time control flowchart.

#### **5. Use Case and Price Mechanism**

The system specifications can be found in Table 1 above. The proposed optimal control scheme can be applied to any building with an EV, solar panels, heat pump, and BES. In this study, a residential building has been chosen as a use case. The building appliance load is obtained from a Dutch distribution system operator and represents a building with a medium to high demand. Both heating and appliance loads are assumed to be non-flexible such that no compromise is given to the user. The power profile of the heat pump is interpolated hourly data obtained from a study performed by the Dutch Organization for Applied Scientific Research [45]. Here, a medium isolated residential building is used with an inside temperature setpoint that varies between 18 ◦C and 20.5 ◦C during night and day, respectively. It is assumed that a time-varying electricity price is obtained from the SGO. The price signal is equal to the Amsterdam Power Exchange market [38,50] only averaged around 0.20 e/kWh, see Figure 6. Additionally, up/downregulation prices *<sup>λ</sup>up*/*down* are obtained from the SGO and shown in Figure 7. The model has been formulated such that it can operate with any kind of regulation market. In this case, the German primary frequency control market was chosen because it resembles the Dutch frequency control market. The German Frequency Regulation market is a symmetrical market, so both up- and downregulation are equally priced. Figure 7 displays the prices obtained for every week in 2018 [51].

**Figure 6.** Electricity price based on the 2018 Amsterdam Power Exchange (APX) spot market, averaged around 0.2 e/kWh.

**Figure 7.** Up- and downregulation prices of the 2018 German market.

#### **6. Results and Discussion**

The results for summer and winter days are shown in Figures 8–11, respectively. Here, the electricity buying price and forecasted and actual PV/load powers are included as well. From both Figures 8 and 10, it is clear that the optimal charging stage trades energy in between times of high and low prices. Similarly, it decides when to store PV energy and when to feed it back to the grid. During winter, the energy demand is much higher, mainly because of the increased heating demand. Additionally, the PV production is much lower and, therefore, less energy is fed to the grid. Without an appropriate control scheme, the BES would only be used in case of excess PV energy, therefore resulting in poor utilization of the BES. However, the proposed model still fully utilizes the available BES capacity in order to reduce the cost of energy. This is also shown in Figures 9 and 11. The EV is charged at night above the required departure charge of 50 kWh, such that it can utilize V2G when prices are high in the morning and again in the evening up to the point where prices are low again at the end of the day. Note that the usable EV capacity is not completely utilized, as the inverter power rating limits the possible power exchanged with the grid, whereas outside the instances of peak/valley prices, the difference between feed-in and retail price is probably not enough to overcome the additional losses caused by the degradation of charging at higher powers. The role of the real-time control scheme can be seen in Figure 8; as PV forecasting errors occur, the algorithm decides whether to use the BES, EV, or grid to compensate for these errors. The grid is used for positive errors (excess of power) and high prices, while the BES/EV is used when errors are positive but prices are low (if possible to store energy) and vice versa for high prices and negative errors. Finally, the effect of the degradation model is seen by the peak powers of the EV/BES. Both are rated at 10 kW. However, their powers only exceed the [−6 kW, 6 kW] range at 3.44% of the time (at 1-min resolution).

**Figure 8.** (left axis:) Optimized power flows for a summer day and (right axis:) energy buying price.

**Figure 9.** Resulting charge from optimized power flows inside electric vehicle and stationary storage for a summer day.

**Figure 10.** (left axis:) Optimized power flows for a winter day and (right axis:) energy buying price.

**Figure 11.** Resulting charge from optimized power flows inside electric vehicle and stationary storage for a winter day.
