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∈

× × + , × ×

(58)

where ⊂ represents the set of regions of all regions that contain household load profiles, and is the set of households in region . Similar to Eq. (35), _, is the price for purchasing electricity from the grid (i.e., the electricity price, V.A.T., energy tax, distribution grid fee), and _, is the price for selling electricity to the grid (the electricity price plus reimbursement from the distribution grid owner) in region at time . Furthermore, _{, ,} is the amount of electricity purchased from the grid, and _{, ,} is the amount of electricity sold to the grid in region by household ℎ at time . For the investments, _, is the size of the battery investment, _, is the size of the PV-panel investment, and _, is the size of the inverter investment in region made by household ℎ. In addition, is the cost of the battery, is the cost of the PV-panel, is the cost of the inverter, and , , and are the annuity factors for the battery, PV-panel, and inverter, respectively. The annuity factor is defined according to Eq. (36). The variables and parameters in Eqs. (52–57) now also operate over the set .

4.5 Energy system models

Two energy system models are used in conjunction with the models presented above, both of which are partial equilibrium optimization models, in that they only model the electricity sector of the economy. The ELIN model is a bottom up, long-term, dynamic optimization model that optimizes the investments in the power sector for the EU-27 countries, as well as Norway and Switzerland. The modeled countries are divided into 50 regions (Fig. 6). This division is based on the current bottlenecks in the European electricity system. The model has a time horizon of Years 2010–2050, with investment decisions being made each year. For each year, there is an intra-annual time resolution of 16 time-steps, representing day and night for weekdays and weekends for four seasons. For a more detailed description of the ELIN model, see Odenberger et al. [77] and Göransson et al. [78]. The results for a given scenario from the ELIN model are used as the input to the electricity system dispatch model EPOD. EPOD takes the description of the power system from a selected ELIN year and carries out optimization, in order to find the least-cost hourly dispatch of the system over 1 year for all regions in Figure 7. For a more detailed description of EPOD, see Unger and Odenberger [79], Göransson et al. [78], and Goop et al. [80]. Both the electric space heating dispatch model and the PV-battery model are connected to EPOD. The Electric space heating dispatch model operates over the Swedish regions SE1, SE2, SE3, and SE4 and the PV-battery model operates over regions SE1 and SE2. The modifications applied to the EPOD model arising from these connections are described below.

35

Figure 7. Regional divisions used in the EPOD and ELIN models.

The two-zone electric space heating dispatch model is included in the system for which EPOD optimizes the dispatch, i.e., EPOD has control over the operation of the heating equipment in the dwellings (Paper V). As a consequence of the inclusion of the electric space heating model, some of the equations and input data in EPOD are modified. The energy balance constraint is changed to:

36

where _, is the electricity demand in region at time-step , excluding the electric space heating demand in ⊂ , where is the set of Swedish regions. In addition, , is the power generated in plant at time-step , and _{, ,} is the electricity traded between regions and at time-step . The set is all power plants in region , and is the set of all dwellings in region . To capture the impact on the overall electricity demand of changes in the dispatch of the heating equipment, a baseline electric space heating demand is defined. Once calculated, it is scaled up and subtracted from the electricity demand profile in EPOD, i.e., as stated, the parameter _, is the electricity demand with the baseline electric space heating demand subtracted in region at time-step for the set .

The extent of the modification of the objective function in EPOD is dependent on whether the fixed interval method is used for the allowed indoor temperature change in the dwellings, as in Eqs.

(14) and (15), or on whether the deviation cost method is used, as in Eqs. (27–31). If the fixed interval method is used there is no change to the objective function, as the change in total cost is captured by changes in the demand and thereby, changes in the electricity generation. For the case in which the deviation cost method is introduced, this penalty cost must be included in the objective function, as follows: part-load costs) for plant at time-step . The introduced variable _, , is the cost penalty for deviating from the set-point temperature for dwelling ℎ at time-step .

The variable _, is in turn subject to:

, = _, × + _, × + _, × + _, × ∀ ℎ ∈ , ∈ (61)

The PV-battery model, Eqs. (52–58), is iteratively optimized together with EPOD (Paper III).

This approach is used so as to capture the feedback effect between the households that are optimizing their investments and EPOD. The iterative procedure is as follows:

1. The electricity system composition is generated in ELIN. The system composition for 1 year is extracted from ELIN and fed into EPOD.

2. EPOD optimizes the dispatch of the system over 1 year, generating a marginal cost of electricity (market price of electricity) for each region and time-step.

3. The marginal cost of electricity is used as electricity prices in the PV-battery investment model. This model then optimizes the investment levels of PV-panels and batteries for each household, as well as the dispatch of the batteries.

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4. The new household net loads [see Eq. (62)] are scaled up to represent all households in the regions.

5. The demand/load curve in EPOD is changed with regard to the new household net load.

6. Steps 2 to 5 are repeated until the convergence criteria or the maximum number of iterations are reached.

The households’ net load is:

, = , − , + , − , × ×

∈

∀ ∈ , ∈ (62)

which is simply the households’ total load, _, , minus the PV-generated electricity plus the energy added to the energy storage minus the energy removed from storage multiplied by the weighting of each household.