The focus of this paper is the development and evaluation of simplified distribution grid models based on a cellular approach. Therefore, this section presents the necessary methodology and discusses the accuracy of the model. Furthermore, the determination of load and production profiles and the determination of synthetic charging load profiles required in the case study are explained.
2.1. Development of Simplified Distribution Grid Models Based on a Cellular Approach
The applied cellular approach described in this manuscript is based on a method designed for multi energy system load flow calculation (gas, heat, electricity) [
35,
38]. A major advantage of the method shown here is the possibility of grid simplifications, decreasing the computation time and therefore allowing an increased time resolution and/or simulation time frames for load flow calculations. Nevertheless, it has to be mentioned, that simplifications always represent a compromise between accuracy and calculation time [
38,
39,
40].
The development of simplified distribution grid models with the cellular approach consists of five steps.
Figure 1 schematically shows this development using one voltage level, which can be easily expanded to multiple voltage levels by repeating steps 2 to 4 separately for each voltage level.
In step 1, the cell classification is done according to the grid topology and the topography of the investigated area. This will be explained in more detail using an artificial sample area (
Figure 2a) with
Figure 2b showing the original distribution grid (in the following called reference grid). The cell classification is done by using a geographic information system (GIS) software, QGIS [
41], with focus on the grid topology of the reference grid. In addition to the grid topology, topographic conditions are used to draw the cell boundaries: a river separating an area into different parts is just one example for such “natural” cell boundaries. Concerning the accuracy of the simplified grid model, the grid topology has a way bigger influence than the geographic topology related cell size. Therefore, the size of the cells or the aggregated amounts of energy per cell is not taken into account during this classification step. In order to achieve the highest possible accuracy of the model, the following guidelines have been proven advantageous:
Consideration of grid topology before geographic topology.
Resulting grid model should correspond to a radial grid.
Prevention of closed ring structure (use of lines with normally open points to connect two cells with each other).
Existing rings in the reference grid should be combined in one cell.
The resulting cell classification of the artificial sample area is shown in
Figure 2c.
In step 2, the cell allocation is done for each cell and voltage level in the power grid. This means that the electrical equipment (transformers, lines, grid nodes, busbars etc.) is allocated to the cells, based on cell classification performed in step 1. In addition, the amount of energy, available measurement (e.g., the load profiles of the individual consumers), and producers or storage units are identified and assigned to the cells.
In step 3, the assigned consumer, producers, and storage units within a cell are separately aggregated into an energy node (
Figure 3). This means, for example, that the amounts of energy of the consumers within a cell are added up to a total energy amount. Available measurement data (e.g., load profile for a hydropower plant), which have already been measured in advance or are provided by energy suppliers, companies etc., or time resolved load profiles are also aggregated into these energy nodes. The energy nodes are located in the geographical cell centre and correspond to virtual busbars for the grid model.
In step 4, the cell-based power grid model is designed using the commercially available software NEPLAN 5.5.6 [
42]. The software NEPLAN is a high end-power system analysis tool for electricity, gas, water, and district heating networks. It is used in transmission, distribution, generation, industry, renewable energy systems and smart grid applications. If this grid model contains more than one voltage level, the following sub-steps must be repeated for each voltage level. First, for each energy node a busbar is inserted into the grid model. Second, the electrical equipment (transformers, consumers, generators, storage units, etc.) and their corresponding time-resolved profile of each cell is integrated into the grid model and connected to the associated busbars. Third, the busbars are connected according to the grid topology of the reference grid. As a result, the cells are connected with each other, allowing power export or import across their cell boundaries. Such corresponding lines are called “interconnectors”. Lines within a cell in the reference grid structure are not part of the simplified grid model and are therefore not inserted in the model as lines. These elements are called “missing elements”. “Interconnectors” and “missing elements” of a schematic distribution grid are shown in
Figure 4a.
Figure 4b illustrates the energy nodes of the geographically located power grid model and their connection according to the reference grid.
As a result of the procedure of the interconnector implementation, connections may occur in the simplified grid model that are not existent in the reference grid (see
Figure 5). The development of the simplified grid model based on the cell division of the reference grid of
Figure 5a shows that lines L1 and L2 are connected in the grid model by the energy node (see
Figure 5b). Since these are not interconnected in the reference grid, load flow shifts would occur between the reference grid and the grid model, resulting in inaccuracies. To counteract these inaccuracies,
Figure 5c suggests choosing a new cell boundary to prevent these connections from developing. As shown in
Figure 5d, the new cell boundary divides the original cell into two cells and the lines L1 and L2 are no longer connected.
Figure 6 shows the reference grid used as a basis for demonstrating the capability of this approach to investigate the influence of increasing e-mobility on a simplified cell-based grid model. This grid is based on an actual 5 kV power grid of a medium sized Austrian city. The 148 nodes of the reference grid are aggregated into nine energy nodes following the previously described procedure. The resulting simplified grid model is shown in
Figure 7. Due to the reduced number of nodes in the model, the duration of the load flow calculations can be reduced by at least a factor seven compared to the reference grid.
In addition to the best possible choice of cell division (
Figure 5), the effects of the “missing elements” must also be discussed. Neglecting the “missing elements” can lead to high deviations between the load flows of the reference grid and those of the simplified grid model. These deviations are mainly caused by losses in “missing elements”. The active power to be transmitted in the line leads to ohmic losses due to the line resistance. These ohmic losses account for less than 1% of the active power to be transmitted. As a result, the active power losses of the “missing elements” have a very small influence on the accuracy of the load flows in the “interconnectors”. Therefore, the accuracy of the active power flows can be regarded as sufficiently accurate, even without consideration of the ohmic losses in the “missing elements”. The reactive power losses caused by the line inductance and capacitance usually correspond to the reactive power to be transmitted in the line. Therefore, neglecting the reactive power losses of the “missing elements” leads to significant errors in the load flows of the “interconnectors”. To reduce these errors, the capacitive and inductive losses generated in the “missing elements” have to be considered in the model.
The losses occurring in a line are defined by the line parameters (active and reactive resistances) and depend on the power (voltage and current) to be transmitted. Therefore, for the consideration of the reactive power losses of the “missing elements”, it is necessary to use elements that are also load dependent. In addition, it must be an element that can be attached to the respective energy node of the cell, since the compensation of load flows can only be affected via parameters of the nodes. This is based on the fact that the abort criterion of a load flow calculation according to the Newton-Raphson method represents the drop below an error limit for the deviations between the previously known node powers and the node powers calculated in the iteration. This means that by adding additional loads in the energy nodes, the reactive power losses of the “missing elements” can be combined independent of their topology and considered for further load flow calculations in a substitution element. For these reasons, E-RLC modules were implemented in the grid model.
To determine the parameters of the E-RLC module, the reactive power losses of the “missing elements” are determined by a load flow calculation of the reference grid and aggregated at cell level:
The aggregated reactive power losses of the “missing elements” (
) is used to determine the capacitive or inductive reactance (
) by:
where
is the nominal voltage of the busbar. The capacitance (
) or inductivity (
of each cell is calculated by:
and
Then, E-RLC modules for each cell are inserted into the grid model and connected to the corresponding busbar. Each module consists of a resistance (
R), an inductivity (
), and a capacitance (
) against ground (
E) in series (
Figure 8), with
and
being the determined capacitance and inductivity of the corresponding cell, respectively. Since the active power losses of the missing elements are not taken into account, the resistance values in the E-RLC module are set to 0.
Although the inductance of the E-RLC module is integrated into the grid model as a parallel parameter instead of a series parameter (substitute circuit: line), the increase in the accuracy of the model can be explained by the lower influence of the inductance on the line losses compared to the capacitance. This lower influence can be explained by the relationship between reactive power and inductance or capacitance. As can be seen from Equations (5) and (6), the inductance is directly proportional to the voltage, while the capacitance is indirectly proportional to the voltage.
By considering the capacitive or inductive losses of the “missing elements” within the cells, the accuracy of the reactive power in the “interconnectors” as well as the accuracy of the utilisation on these lines is increased. The accuracy improvement of the reactive power flows for selected “interconnectors” shown in
Figure 9 was achieved by integrating the E-RLC modules. These results are based on load flow calculations for maximum load.
Comparing the deviation of the reactive power load flow for different lines with and without consideration of the reactive power losses of the “missing elements” (
Table 1) shows that the implementation of E-RLC modules significantly reduces most deviations (
Figure 9 and
Table 1). Although all errors can be reduced, this model is not suitable for analyses of voltage drops at the grid nodes due to the remaining deviations of the reactive power load flows in some lines, and based on the fact that the reactive power has influence on the node voltage. Furthermore, the energy node is an aggregated grid node that contains the sum of all consumers and producers within a cell and has nothing in common with a real node from the reference grid. It has to be mentioned that the amount of “faulty lines” scales with the complexity and size of the power grid.
In step 5, the accuracy of the simplified model is analysed in two sub-steps. At first, the software NEPLAN is used to perform load flow calculations using the Newton-Raphson method for the highest possible load in the reference grid as well as for the cell-based grid model. In this first sub-step, the accuracy of the overall system is checked by comparing the slack bus of the grid model with the reference grid.
Table 2 shows this comparison for the imported active power and reactive power as well as the active power and reactive power generated in the grid for the slack bus, based on data for the operating point of the maximum load. It can be seen that the relative deviations for imported and generated power is less than 2%. The deviations of the imported active and reactive power is mainly attributed to the deviations of the load flows in the lines. In total, these deviations influence the imported active and reactive power and lead to relative deviations of 0.89% and 1.60%.
In the second sub-step, the active and reactive power flow deviations between the grid model and the reference grid for the “interconnectors” and the transformers are compared. As already mentioned, the E-RCL modules can be used to reduce the deviations of the reactive power flows between the grid model and the reference grid, but these are still very high in individual cases (relative deviations of up to 200%).
Figure 10 shows the absolute and relative deviations of the active power flows between grid model and reference grid. It can be seen that two lines have relative deviations of >50%, while the remaining lines have relative deviations of <15%. The four lines with the highest absolute deviations are L35_1, L37_1, L62_1, and L63_1.
To analyse the high deviations seen in lines L62_1 and L63_1, their position in the grid model and the reference grid must first be localised. As shown in
Figure 7, the affected lines form a closed ring with the L23_1 line. This is the only closed ring in the grid model. While the active power load flow of line L62_1 in the grid model is smaller to the corresponding line in the reference grid, the active power load flow in line L63_1 is larger. Therefore, a load shift occurs due to the ring topology in the grid model. Although, the relative deviation is high in lines L62_1 and L63_1, absolute errors are in the same order of magnitude as other lines (see
Table 3). While the active power flows for lines L35_1 and L37_1 are between 750 and 1500 kW, those of lines L62_1 and L63_1 are below 100 kW. This difference in the active power flow results in high relative deviations of 56% and 54% for the lines L62_1 and L63_1, respectively. For the lines, L35_1 and L37_1 the relative deviations are only 2% and 3%, respectively. For this reason, care should be taken during cell division to avoid ring structures in the grid model. The highest accuracy of the load flows in the grid model is achieved if the topology corresponds to a radial grid.
The deviation in lines L62_1 and L63_1 should be kept in mind when using the model to determine the increasing grid load with the integration of future e-mobility. As the simplified model always represents a compromise between accuracy and calculation time, no clear limits can be set to the accuracy of the model. Therefore, the accuracy of the model depends strongly on the desired application. If the defined accuracy of the model cannot be achieved, a detailed check is carried out and possible reasons for the deviations are investigated. The optimisation potential is examined with regard to changes in cell boundaries. However, this accuracy check is repeated until the desired accuracy is achieved.
Following steps 1 to 5 results in a simplified cell-based grid model representing the actual power grid and allows for time-resolved load flow calculations with annual load profiles in a short time. Once the model has been developed, it is possible to integrate electrical production potentials, charging stations for e-mobility and other elements that allow flexible use (heat pumps, power to heat systems, etc.). In addition to energy analysis (e.g., self-coverage), this approach also allows the determination of cells that are susceptible for grid instabilities. These cells can be analysed for the grid-related impact of a specific technology.
2.3. Determination of Synthetic Charging Load Profiles of Electric Vehicles
To analyse the impact of a large amount of EVs, the development of synthetic charging load profiles is necessary. The presented methodology for the development of this charging load profiles essentially consists of four steps
Figure 12: (1) data preparation, (2) determination of cell and user group specific charging steps, (3) modelling of the charging curve, and (4) aggregation of the load profile.
In step 1, the database for the determination of synthetic load profiles is created based on registration statistics, traffic analysis and mobility patterns. From electric vehicle types registered in Germany [
48], a distribution function of the EV type is derived. Each EV type is allocated by a battery capacity and an average EV energy consumption (kWh/100 km) depending on the season. The required definition of the seasons corresponds to that of the standard load profiles provided by BDEW [
43].
The statistical data from traffic analysis being used to calculate load profiles in the presented approach contain a wide range of different parameters (average duration of stay, distance travelled, number of trips, etc.), which depend on the topography of the investigated area (suburbs, city centre). Additionally, each trip can be associated with one of seven pre-defined trip purposes, referred to as user groups. A distinction is made between a trip home, a trip to work with private or official car, a trip for shopping, trip for execution (e.g., doctor’s visit), trip to leisure activities, and a trip to education. For further use, a distribution function over the distance travelled and the number of trips per weekday is determined for each cell and user group.
The mobility pattern contains empirical factors that are used based on the number of trips per weekday to determine the number of trips for Saturday and Sunday for each cell and user group. The number of trips per day corresponds to today’s mobility behaviour, meaning that in a scenario with an EV penetration of 100%, each distance travelled represents a trip with an electric vehicle without taking into account any change in mobility behaviour. Furthermore, the mobility pattern includes original destination matrices according to Bosserhoff [
48]. These matrices are based on the statistical evaluation of traffic behaviour and are available as daily distributions for 15 various trip purposes. These distributions describe the relative share of arriving and departing vehicles in the total vehicle volume of a day per day hour. While for the purpose “trip home”, a daily distribution for arrival and departure exists and can be assigned directly to the user group “trip home” from the traffic analyses, which is not possible for the user group “trip to work”, for example. The user group “trip to work”, regardless of whether it is a private car or an official car, is strongly dependent on the workplaces available in the cell, more precisely on the start and end of the respective work. This means that a “new” daily distribution is aggregated for this user group on the basis of different Bosserhoff daily distributions and using a distribution of existing workplaces within the respective cell. As a result, an individual daily distribution of arriving and departing vehicles will be obtained for each cell for the user group “trip to work”. For these daily distributions and those Bosserhoff daily distributions that can be directly assigned to a user group, cumulative distribution functions of arriving and departing vehicles are finally generated for each cell. This cumulative distribution functions are used in the probabilistic approach in step 2.
Figure 13 shows two examples of original destination matrices according to Bosserhoff for the purpose “trip home” and “trip to work-shift operation” as well as the cumulative distributions determined.
Following the preparation of the traffic analysis and the mobility behaviour, the EV penetration and the simulation period is determined in step 2. Furthermore, the arrival and departure time, the distance travelled, the EV type, and the charging strategy are determined for each charging process within a cell and user group. Due to the stochastic nature of the mobility behaviour, the determination of the distance travelled, arrival time (=start charging process), departure time, and EV type is performed via a probabilistic approach using a random number generator, similar to references [
31,
50,
51]. Another important parameter is the charging strategy. To study the impact of different charging strategies, we define three different options, which can be selected separately for each user group.
In charging strategy 1, the charging power is selected separately for each user group. All charging processes within a user group thus charge with the same charging power. For example, as is quite common with today’s charging stations, charging powers can vary between 3.7 and 50 kW.
For charging strategy 2, the charging power is determined for each charging process based on a probability distribution function. This distribution function can, for example, be determined from the data of charging stations and therefore describes how many users have charged with which charging power.
Charging strategy 3 simulates the possibility of reduced charging power under consideration of the available charging time, corresponding to the duration of stay. The determination of the reduced charging power takes place in step 3. Since each loading process is simulated independently of the previous or subsequent one, it is not possible at this point to shift the charging process within the duration of stay as part of this strategy. The influence on the time shift would have to be taken into account when preparing the distributions for arrival and departure times.
After the parameters have been determined, they are written to a log file, which is used in step 3. This process is repeated for each day in the selected simulation period based on the number of trips per day and the selected EV penetration within a cell and user group. After the last repetition, this log is saved for each cell and user group.
In step 3, the charging curve is modelled for each charging process of each cell and user. For this, the duration of stay is calculated using the arrival and departure time, to establish whether the battery can be fully charged. Additionally, the necessary charging energy is determined based on the distance travelled and the average EV consumption (kWh/100 km). The battery capacity and the necessary energy to charge the battery are used to determine the state of charge at the end of the trip, i.e., at the beginning of the charging process. Next, the duration of the charging process can be determined according to the charging strategy and the respective charging power. In case of charging strategy 3, the corresponding charging process is first defined by a constant charging power. Based on this charging power, the duration of the charging process is determined. Afterwards, a comparison is made between the duration of stay and the duration of the corresponding charging process, see
Figure 14. If the duration of the stay is shorter than the duration of the charging process, the charging curve is calculated by using the constant charging power. If the duration of the stay is longer than the duration of the charging process, the charging power is adjusted to the lowest possible value at which the consumed energy of the last trip can be recharged for the corresponding duration of stay.
The state of charge at the end of the trip and the charging power are the basis for modelling the charging curve. As described in references [
52,
53], constant current (CC)—constant voltage (CV) charging is applied. This means the battery is first charged with a constant current (i.e., constant power (
Pconst)). At the changeover point (
s = 80% SOC) constant voltage charging follows. In this charging phase, the charging current decreases automatically. The charging power as a function of the state of charge (
SOC) is therefore calculated by Equation (7): [
52]
where,
kL is the correction factor, which considers the nominal battery capacity and the switch off of the charging power. It has to be mentioned that Equation (7) is only valid for lithium-ion batteries, which are widely used in EVs.
In step 4, all charging curves within a cell and user group are aggregated to one synthetic load profile for the respective cell. In a first stub-step, all charging processes within a user group are added up to a sum load profile resulting in charging curves for each user group and cell.
Figure 15b shows an aggregated charging curve for one user group, which are illustrated in
Figure 15a (for illustration purposes, only 10 charging processes are shown). In a second sub-step, the aggregation of all charging curves at user group level within a cell are aggregated to a synthetic load profile. In
Figure 16c, the aggregated charging curves of all charging processes within a cell for each user group from sub-step 1 are used for a selected cell to represent sub-step 2. Depending on the user group, these aggregated charging curves contain between 100 and 500 charging processes per day, and therefore the characteristic from
Figure 16b is no longer recognisable.
Figure 15d shows the example of the aggregation of all charging curves at user group level from
Figure 15c.