A Voltage Sensitivity Based Equivalent for Active Distribution Networks Containing Grid Forming Converters

The evolution of load-dominated to active distribution networks with significant dispersed generation calls for accurate models of these networks in the context of system stability analysis. Embedding detailed distribution network models in a dynamic transmission system model is impracticable due to model complexity. Complexity-reduced equivalent dynamic models solve this obstacle. However, equivalent models that consider grid forming converters emulating inertia have not been addressed in research yet. To fill this gap, this work proposes a novel gray-box method for aggregating active distribution networks containing grid forming converters. The approach utilizes voltage sensitivities to represent the grid’s strength at the grid forming converter’s connection point. We compare the proposed method with an existing gray-box approach capable of creating equivalent models for networks dominated by conventional grid following converters. Both aggregation approaches are applied on an active distribution network model. Simulation results for different events of the detailed and both equivalent models are compared. The equivalent model aggregated with the proposed approach reproduces the detailed network’s dynamic behavior adequately, while the existing approach fails to meet validation criteria. Hence, the proposed method provides an equivalent model that is capable of substituting detailed distribution network models in stability studies of renewable power systems.

to a renewable system, characterized by converter based generation (CBG) in the distribution system. In a conventional power system distribution networks behave quite uniformly and can be adequately modeled by rather simple models, e.g., as equivalent loads. However, a more complex, ideally detailed, modeling is necessary as the penetration of generation in general and CBG in particular in distribution networks increases. Otherwise, the influence of these active elements within active distribution networks (ADN) on the transmission system cannot be captured adequately. To avoid detailed ADN models as parts of system models, the concept of an equivalent dynamic ADN model (EDAM) was developed [1], [2], [3], [4]. An EDAM reproduces the significant dynamic behavior of the detailed ADN in complexity-reduced models.
With the increasing penetration of CBG, new challenges for converter control emerge. Phasing out synchronous machines providing inertia jeopardizes the system stability. Grid forming converters (GFMC) emulating inertia of synchronous machines are expected to become essential components of stable CBG dominated systems [5]. This increasing importance leads to the necessity of considering the GFMC's dynamic behavior in stability analysis of future power systems.
Summing up, CBG dominated distribution networks affect the way in which system stability is studied. Accurate EDAM as parts of system models for comprehensive stability analysis and the contribution of GFMC to system stability become more important. The combination of these two aspects displays a research gap not addressed in previous work, i.e., the consideration of GFMC in EDAM.

B. Literature Review
This section classifies state-of-the-art methods for creating EDAM and highlights the research gap. Moreover, we introduce relevant converter control strategies. Since the proposed method utilizes voltage sensitivities in a network, the derivation and interpretation of these are elaborated.
1) Equivalent Dynamic ADN Models: Detailed modeling of ADN, for use in network models for stability studies, requires high computational efforts and data availability. Moreover, for most stability studies the transmission system is of interest, i.e., events located here are simulated and its stability is evaluated. Aggregating a detailed ADN model to an EDAM resolves the mentioned obstacles, but care is to be taken in order not to neglect significant features. An EDAM This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ represents the ADN and its behavior in a lower order model, i.e., a less complex mathematical representation compared to the detailed model. Therefore, the use of EDAM allows manageable comprehensive stability studies of renewable power systems with CBG in the distribution networks.
Motivated by [6], [7], [8], [9], a classification of methods for creating EDAM is proposed in previous work [10] and shown in Figure 1. Here, the different generation mix of a detailed network model to be aggregated distinguishes the methods: ADN with conventional generation only, i.e., synchronous generators, and ADN with a relevant penetration of CBG.
Most commonly used methods for creating EDAM of networks with conventional generation only are coherency and modal based approaches [11]. Coherency based EDAM are obtained by aggregating coherent generators. A similar rotor angle swing qualifies generators to be aggregated applying Zhukov's method [12]. With the method developed by Dimo [4], the whole network can then be reduced [13]. As CBG lacks any rotor angle, this method is not applicable for CBG dominated ADN.
Modal based aggregation approaches reduce the order of the detailed system by linearizing it and focusing on dominant eigenvalues. As such, only the small signal behavior can be replicated adequately [3]. However, for stability analysis the simulation of severe faults leading to nonlinear large-signal responses of CBG is essential [9].
The majority of promising methods for creating EDAM of CBG dominated systems are parameter identification approaches [14], which utilize data as input for the parameter identification algorithm. Data can be gained either from real networks or from detailed simulation models, where the latter case outlines data generation for the purpose of data exchange. With the gained input data, parameters of either a black-box or a gray-box model can be identified.
Black-box based EDAM are created by training an artificial neural network. Input training data can comprise time series of loads, CBG, and switching states. The parameters of the artificial neural network are optimized to match the input data to the corresponding output training data, i.e., power exchange at the boundary bus, as accurately as possible [15]. Black-box based EDAM can be found in [15], [16].
Methods for creating gray-box model based EDAM can be further categorized into generic model based and clustering based approaches. The former utilize a generic model structure often comprising a ZIP load and an induction motor model [17], [18]. This model can be extended by a CBG or an exponential recovery model [19], possibly adapted to take generation into account [20]. The model parameters, including the dynamic control parameters, are estimated by utilizing parameter identification algorithms based on comprehensive input data such as time series of voltage, frequency, active and reactive power [21]. Examples of generic model based gray-box EDAM are published in [21], [22], [23], [24], [25], [26], [27], [28], [29].
Both black-box and generic model based gray-box methods don't rely on insight into the network to be aggregated. However, in case input data for parameter identification are gained from detailed simulation models, the computational effort for data generation is high. Also, data from real network measurements need to be processed and mostly don't comprise large disturbances. Further drawbacks are the dependency of the created model on the data set used for the parameterization and the high computational effort of parameterizing such an EDAM.
Clustering based methods are particularly suitable if detailed information on the ADN is available. The goal of this work is to provide EDAM for stability analysis of a future system. As opposed to modeling real networks, it can be sufficient for stability analysis to model representative expected ADN structures from which EDAM will be derived. Therefore, full insight into the network to be aggregated, i.e., the ADN, can be assumed. In this case, clustering based methods can be applied to utilize this knowledge.
An overview of international industry practice to derive clustering based gray-box EDAM in a simulation environment is given by [30], [31]. Netting the loads and generators to a single equivalent load is the most commonly used method (Netting Approach) [32]. A further more detailed model developed by the Western Electricity Coordinating Council (WECC) includes, besides the equivalent load, an equivalent transformer, impedance and generator (WECC Approach) [33].
These approaches lack a differentiation between generation technologies, control strategies and voltage levels leading to insufficient simulation results for networks with high CBG penetration [14]. This drawback is addressed in [34] with the introduction of the Technology-Control-Clustered Approach (TCA). Based on knowledge about the network to be aggregated, i.e., technologies and control strategies of components as well as static load flow data, the TCA application creates an EDAM capable of reproducing the dynamic response of a CBG dominated ADN.
TCA is a gray-box aggregation method that clusters the detailed network's components according to technology, control strategy and voltage level [34]. The components are aggregated as a single equivalent unit based on the respective cluster. Each voltage level has a single equivalent node, where the corresponding components are connected to. Equivalent transformers and equivalent impedances connect the different voltage levels. A boundary bus is the common link between the transmission system and the ADN. For optimizing the equivalent impedance parameters, the objective is to minimize the deviation of the EDAM's steady-state power flow at the boundary bus from the power flow observed at the boundary bus of the detailed network model.
However, in the EDAM created with the TCA, all network components of one voltage level are connected to a single busbar. Therefore, this EDAM cannot capture the electrical grid's strength at each node of the detailed network as well as the electrical distance from each component in the network to the fault location. As the dynamic response of GFMC highly depend on those two factors, the dynamic behavior of the detailed ADN differs significantly from its EDAM as soon as GFMC are introduced [35].
2) Converter Control Strategies: ENTSO-E has introduced three classes of converter control depending on their respective capabilities. Class 1 refers to the basic level focusing on converter survivability, while class 2 includes more advanced control. In this work both classes are defined as grid following converters (GFLC). GFMC capable of providing inertia are listed as class 3 [36]. a) Grid following converters: The most widely used converters, e.g., for application in PV generators, behave approximately like a current source. The desired active and reactive power is fed by changing the converter's output voltage depending on the voltage at the PCC on a very short timescale. Typically a Phase-Lock-Loop (PLL) and a current control loop are applied. An external voltage source is needed to provide the reference values for voltage and frequency [37]. Hence, GFLC are not capable of working in a stand-alone mode.
A GFLC can behave like a constant current source controlled to specific active and reactive power set points. Or it can be operated as a controlled current source, e.g., voltagedependent reactive power output, capable of dynamic grid support. Being more advanced, ENTSO-E attributes converters with such a control to the Class 2 Power Park Modules [36] and this work focuses on that concept for GFLC. b) Grid forming converters: By acting as a voltage source, GFMC are able to work in stand-alone mode and to provide inertia to the system. As opposed to synchronous machines tolerating short-term overload, the GFMC's output current is strictly limited to protect the converter's power electronics. Such a control represents the future needs of a system with high penetration of CBG and the ENTSO-E refers to it as the Class 3 Power Park Modules [36].
For an application as a network's sole generator, e.g., in islanded networks, GFMC can be operated as a constant voltage source with fixed frequency and voltage set points. Parallel operation of multiple GFMC, e.g., in interconnected networks or microgrids, demands a behavior as a controlled voltage source, whose output voltage can change with a slow dynamic response. The latter kind of GFMC control is implemented in this work. GFMC development and modeling is the focus of current research work and multiple approaches for GFMC control are possible. Droop control [37], [38], [39], [40], virtual synchronous machine [41], [42], [43], [44] or virtual oscillators [45] are some examples of possible realizations of GFMC control. Moreover, a combination of modeling approaches has been developed, e.g., a droop based virtual synchronous machine [46]. It is important to mention that some control concepts are very similar and partly equivalent. A comparison study of different control concepts can be found in [47]. In this paper, an electromechanical model of a droop based GFMC according to [48] is applied. The model is capable of the mentioned current limitation. An evaluation of the impact of other GFMC concepts is presented in Appendix B.
3) Voltage Sensitivities in a Network: The voltage sensitivity of a node is defined as the change in voltage magnitude or voltage angle due to a change in active or reactive power at that specific node. The inverse of the Jacobian matrix, which is used in the Newton-Raphson power flow calculation method, can be interpreted as a sensitivity matrix, from which voltage sensitivities can be extracted. The calculation of the Jacobian matrix is briefly explained in the following paragraph.
The active power P and reactive power Q balances at node i can be written as where Y ij = G ij + jB ij are the bus admittances of the network [49]. V i and ϑ i are the voltage phasor magnitude and angle, respectively. ϑ ij = ϑ i − ϑ j defines the voltage angle difference between node i and j.
In order to derive voltage sensitivities, we linearize equations (1a) and (1b) by a first order Taylor series approximation. Assuming that node 1 is the slack node, the resulting set of linear equations can be written in matrix form as where Z is the vector of power deviations, X is the vector of small voltage deviations, and G is the Jacobian matrix, as a matrix of partial derivatives of independent variables. Solving these for X yields The inverse Jacobian matrix F contains the basic information for the voltage sensitivity calculation [49], [50]. It is important to mention that the Jacobian matrix used corresponds to a stationary solution of the load flow equations.
For this work we focus on the diagonal elements of the four submatrices H, N, J, and L of (5). These elements refer to the response at one node to changes at the same node. As such, they describe how sensitive the node voltage is with respect to changes of active and reactive power injection changes. The voltage and ∂ϑ i ∂Q i at node i represent the change in voltage with respect to a change in active or reactive power. For example, a voltage sensitivity ∂ϑ ∂P of 0.5 1 pu corresponds to a voltage angle change of 0.005 as a result to an active power change of 0.01 pu at the same node. The stronger the grid, the lesser the voltage is influenced by power changes. Hence, these diagonal elements are suitable parameters to express the grid's strength at a certain node.
For an exemplary radial medium voltage network, in which the transformer to the high voltage network is placed in the center of the network, the voltage sensitivities ∂ϑ i ∂P i are calculated and plotted as a heat map in Figure 2. It can be seen that voltage sensitivities are low around the transformer, and increase with the increasing distance to the network's center. Since the transformer connects this distribution network to a stronger transmission network (represented by a slack node), the grid is the strongest around the transformer. Hence, voltage sensitivities at nodes around the transformer are the lowest. No voltage controlled node is implemented in the shown network. Such would decrease the voltage sensitivity around its node.
These observations are coherent with the statement that voltage sensitivities are a suitable measure of the grid's strength. One could argue that parameterizing impedances according to the electrical distance between a node and the transmission system would yield the same result. However, while the calculation of the electrical distance is non-trivial, voltage sensitivities can easily be derived from the load flow calculation results. Moreover, if the grid's strength were represented by the electrical distance alone, the effect of voltage controlled nodes would be neglected.

C. Contribution and Paper Organization
Considering GFMC in EDAM is the research gap addressed in this work. The paper proposes a validated methodology, termed as Sensitivity-Technology-Control-Clustered Approach (STCA), to create EDAM of CBG dominated distribution networks including GFMC. Here, to map the GFMC's dynamic behavior as accurately as possible, for each GFMC of the detailed ADN there is a corresponding equivalent GFMC connected to the EDAM with an impedance according to the detailed network's topology. The impedances are tuned dependent on the individual voltage sensitivities ∂V i ∂P i , and ∂ϑ i ∂Q i at the point of common coupling (PCC) of the GFMC i in the detailed network. As a result, the equivalent GFMC has a similar influence on the voltage at the GFMC's PCC compared to its counterpart in the detailed network. Such a consideration of GFMC in the EDAM allows for a proper representation of the detailed ADN even for large signal disturbances leading to nonlinear GFMC behavior.
The proposed STCA is validated by comparing the active and reactive power flow of both detailed model and EDAM (aggregated by STCA) at the boundary bus between the ADN and the transmission system. In this paper, the method is applied to an ADN characterized by photovoltaic (PV) generation as CBG covering almost a 100 % of the ADN's demand. Quasi-steady-state simulations (RMS) are conducted for five different events in order to evaluate the performance of the EDAM.
The main contributions of this paper are the following: 1) Opposed to previous work focusing on the representation of dynamic loads and GFLC, we propose an EDAM derivation methodology that captures the dynamic behavior of GFMC within the ADN, since existing methods do not perform well for ADN that contain a significant amount of GFMC.
2) The proposed STCA utilizes knowledge about the network to be aggregated and is independent of measurement data or dynamic simulation results for the purpose of generating input data for the EDAM parameter identification. Hence, the application of the STCA as a clustering based method requires less computational effort compared to generic model based gray-box methods or black-box methods.
3) The proposed STCA is most suitable for studies on future power systems. 4) A generic EDAM topology is proposed to render the STCA application to different network model structures possible. 5) The comparison of simulation results of both a detailed network model and its corresponding STCA based EDAM for different disturbances shows the validity of the STCA. 6) The performance of a TCA based EDAM is compared with the STCA based EDAM to demonstrate the limitations of state-of-the-art approaches and the contribution of the STCA. 7) Validation of the STCA was conducted for a droop based GFMC, a Synchronverter, and a Virtual Synchronous Machine. 8) Events utilized for validating the STCA comprise balanced and unbalanced faults.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
The proposed STCA for creating EDAM considering GFMC is introduced in Section II. For validating the STCA, a test network is developed and the corresponding EDAM is created. Section III gives an overview of the models used in this work. Also, the simulation results of five different events are described and a numerical validation is performed to evaluate the performance of the proposed method. This work concludes with a discussion of the results in Section IV and a summary with an outlook in Section V.

II. SENSITIVITY-TECHNOLOGY-CONTROL-CLUSTERED APPROACH
The EDAM aggregated by TCA is capable of capturing the dynamic behavior of GFLC dominated networks, but fails to adequately represent GFMC [14]. The proposed STCA addresses these drawbacks by further developing the TCA. The electrical grid's strength at the GFMC's PCC is one important factor for the GFMC's behavior. Also the electrical distance to the fault location, usually being within the transmission system for stability analysis, influences the response of the GFMC. These factors are considered in the STCA based EDAM by connecting each GFMC of the detailed network via optimally parameterized impedances to the EDAM dependent on the detailed network's topology.
In a first step, all components of the detailed network are assigned to clusters similar to the clustering process in the TCA [34], see also Section I-B1. All loads and GFLC of the detailed network are clustered and aggregated to equivalent components according to the respective clusters. Additionally, all GFMC of the detailed network are individually considered in a separate GFMC cluster. This allows a proper consideration of the voltage sensitivities ∂V i ∂P i , ∂V i ∂Q i , ∂ϑ i ∂P i , and ∂ϑ i ∂Q i at the corresponding PCC.
The gray-box model is then created in a second step, starting with the equivalent transformer. Its parameters are the same as the transformer between the detailed ADN an the transmission system. The other equivalent components are created according to the clusters with one equivalent component per cluster. Active and reactive power demand of an equivalent load is given by the sum of the demand of each load in the respective cluster in the detailed network. Analogously, the sum of active and reactive power generation of the individual GFLC per cluster in the detailed network corresponds to the respective equivalent GFLC generation. Opposed to the model creation of the TCA based EDAM, the equivalent loads and GFLC are connected to the lower voltage side of the equivalent transformer without equivalent impedances.
Parameters of the equivalent GFMC should match the ones of the corresponding GFMC in the detailed network. The equivalent GFMC are connected to the lower voltage side of the equivalent transformer with equivalent impedances. This connection depends on the detailed network's topology. It can be described as a graph G = (N , E) with vertices N i ∈ N and edges E ij ∈ E with i, j ∈ N and i = j. A vertex N i can be either a GFMC with its PCC node and all its corresponding parameters or a branch-off. An edge E ij connecting the vertices N i and N j contains the parameters of an admittance Y ij , that is, The boundary bus has the same properties as a branch-off vertex and is set to be the vertex N 0 , from which edges lead to n subgraphs referring to n branches in the detailed network. These subgraphs are trees, and they can be interconnected via an edge.
The graph is built to represent the GFMC location in the detailed network as closely as possible. Two GFMC in the same branch are two vertices connected with an edge in series. Two GFMC in two different branches not connected will be in two different subgraphs only connected via the vertex N 0 . The number of vertices equals the number of GFMC in the detailed network in addition to the number of branch-offs and the vertex N 0 . The number of edges depends on the number of vertices and interconnections between two subgraphs. The topology of an exemplary STCA based EDAM with two GFMC in one branch and one GFMC in a separate branch can be seen in Figure 4.
The result of the clustering and gray-box model creation steps is a model structure with correct parameters of the equivalent loads and generation components. The remaining unknown parameters of the equivalent impedances are found in the last step of the EDAM derivation. In an iterative procedure, the parameters of the equivalent impedances are set to achieve the same voltage sensitivities at the PCC of each equivalent GFMC i compared to the PCC of the corresponding GFMC in the detailed network. In a final step, a slack load is created and connected to the lower voltage side of the equivalent transformer. It is parameterized with the objective to have the same steady-state active and reactive power flow at the boundary bus in the EDAM compared to the one observed at the boundary bus of the detailed network. In some cases, the slack load can also be operated as a negative load, i.e., a static generator. Figure 3 (c) shows an exemplary EDAM aggregated by STCA. The equivalent loads and equivalent GFLC are clustered similar to the EDAM aggregated by TCA and connected to the lower voltage side of the equivalent transformer without impedances. Here, there are two different load clusters resulting in two equivalent loads. Also, two GFLC control clusters lead to two equivalent GFLC. The GFMC are connected to the lower voltage side of the transformer via impedances according to the network topology. In this case, all GFMC in the detailed network are connected to different branches of the radial ADN, as it can be seen in Figure 3 (a), resulting in a parallel connection of the equivalent GFMC with one GFMC per branch in the EDAM.

III. EVALUATION OF STCA
In this section, the performance of the proposed STCA is evaluated and validated by applying it on a test network model. Five event scenarios are simulated and results are compared between the detailed network and the EDAM aggregated by STCA. Furthermore, the TCA is also applied on the test network model and results are compared for benchmarking purposes.

A. Detailed Grid Model
The test network model is divided in two parts. In all cases, the transmission system is represented by a 230 kV voltage source connected to the boundary bus by a 50 km transmission line. The transmission line parameters are based on the CIGRE benchmark subtransmission network line parameters in the European configuration [51].
The system to be aggregated is a 10 kV ADN based on a network model created with the open-source tool DINGO [52]. DINGO creates synthetic radial medium voltage networks based on publicly available data of German power system networks. The created network is modified in order to obtain a converter dominated ADN. To this end, 99 % of the ADN's total demand, including grid losses, is covered by PV generators. GFMC account for about 60 % of the total PV generation capacity.
The PV systems with grid forming controlled converters are operated with a battery storage to enable an additional power provision for certain events. The remaining 40 % are GFLC implemented with two different dynamic models according to [53]. The generic model for distributed and small as well as for large-scale PV plants differ in their threshold values for disconnection and post-fault power generation. Around 83 % of the total demand is represented by five dynamic composite load models according to [54]. This dynamic composite model comprises three three-phase double-cage induction motors, one one-phase air conditioner performance-based motor, an electronic load, and a static load. The remaining 17 % of the total demand is modeled as constant impedance loads. Figure 3 (a) shows the detailed network model including the transmission system and the ADN. Data for generation and demand of the network model are shown in Table I. The large-scale PV plants are connected centrally at the lower voltage side of the 230 kV/10 kV transformer, while the small

B. Equivalent Grid Model
The detailed ADN model is aggregated according to the proposed STCA. The resulting EDAM is shown in Figure 3 (c). The TCA was also applied to the detailed network model for comparison and benchmarking purposes and is shown in Figure 3 (b). The detailed network is comprised of six GFMC with one GFMC per branch resulting in Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. the same number of equivalent GFMC connected in parallel with corresponding six equivalent impedances in the STCA based EDAM. The GFMC are aggregated to one equivalent component in the TCA based EDAM.
GFLC in the detailed ADN model are implemented with two dynamic control models, i.e., different behavior during events or faults. Hence, the EDAM aggregated by both TCA and STCA are comprised of two equivalent GFLC representing these two control strategies. Also, two different load models, i.e., dynamic composite load and constant impedance load, are applied to represent the ADN's demand. This leads to two load clusters resulting in two equivalent load models for both TCA and STCA based EDAM. All loads and GFLC of each cluster were aggregated to a single equivalent unit by summation of their active and reactive power demand and generation, respectively. In addition to the equivalent loads, a slack load as described in Section II is connected to the lower voltage side of the equivalent transformer of the STCA based EDAM.
As mentioned in Section I-B1, parameter identification methods also comprise generic model based gray-box and black-box approaches. Drawbacks of these approaches are the effort required to generate comprehensive training data for various disturbances and faults and the dependency of the resulting EDAM on these data. Additionally, identifying parameters of the generic gray-box model and training the artificial neural network, respectively, requires high computational resources. Nevertheless, it is expected that such models would pass the validation conducted in Section III-D as has been shown in previous work [15], [16], [21], [22], [23], [24], [25], [26], [27], [28], [29]. However, the proposed STCA requires significantly less computational effort for creating the EDAM by utilizing knowledge about the detailed network model as described in Section II. This depicts an advantage of clustering based gray-box approaches, which is particularly relevant for conducting stability analysis of future power systems in which multiple ADN and generation scenarios are considered. Hence, this work focuses on the comparison of the STCA with a state-of-the-art clustering based gray-box method, i.e., the TCA.

C. Simulation Results
The three network models depicted in Figure 3 are exposed to events in the transmission system. RMS simulation results of the two EDAM aggregated by TCA and STCA are compared to the detailed network model. The simulations are performed with an integration time step of 1 ms using the software DIgSILENT PowerFactory.
Five scenarios are considered to evaluate the proposed STCA in different events. The first event is a phase angle jump from 0 to 10. The second event is a frequency jump from 50 Hz to 50.25 Hz. Both events are induced by the voltage source. The other events are a three-phase, two-phase, and one-phase short circuit fault at the PCC of the voltage source as marked in Figure 3. All events take place at 0 s and the short circuits are cleared after 130 ms. For clarity, the simulation results for the two-phase and one-phase short circuits are not shown in the following figures. Nevertheless, they are considered in the validation in Section III-D.
An important feature of a GFMC is the capability of limiting the output current. The dynamic control model of the GFMC calculates the set point for output current in a synchronous reference frame (dq-frame) based on the droop control output [48]. If this calculated set point exceeds the maximum admissible current, the current set point is limited accordingly. Figure 5 shows the calculated current set points of a single GFMC in the detailed network and the corresponding GFMC in the EDAM aggregated by STCA. Also, the calculated current set point of the equivalent GFMC in the TCA based EDAM is plotted.
For the phase angle jump, the equivalent GFMC of the TCA aggregated EDAM reaches the current limit for a shorter time period compared to the GFMC of the detailed network. The GFMC of all three models do not exceed the maximum current after the frequency jump. For the three-phase short circuit fault, the current limit is reached by the equivalent GFMC of the EDAM aggregated by both TCA and STCA for a similar time period compared to the GFMC of the detailed network. Figure 6 shows the active and reactive power flow from the transmission system to the ADN at the boundary bus of detailed network and EDAM aggregated by TCA and STCA for different events. Before the events occur at 0 s, active and reactive power of both EDAM match those of the detailed network well. This is ensured by the parameterization of the equivalent impedance in the TCA based EDAM and by the parameterization of the slack load in the STCA based EDAM.
The phase angle jump leads to swings of active and reactive power in all models in transition to the pre-event values. It can be observed that the STCA based EDAM captures the dynamic behavior of the detailed ADN better than the TCA based EDAM. The differences in the dynamic behavior become more severe for the frequency jump event. After the event occurs, active and reactive power reach new steadystate values. However, the post-fault reactive power of the EDAM aggregated by TCA is higher compared to the detailed network. The post-event active power of the TCA based EDAM is close to the detailed network, but differences in the dynamic response can be seen. The STCA based EDAM reproduces the dynamic behavior of the detailed network well, for both active and reactive power.  As the most severe event, the three-phase short circuit fault also leads to differences between the dynamic behavior of the TCA based EDAM and detailed network. During the short circuit, the power flows of TCA based EDAM are close to those of the detailed network. After the short circuit is cleared, EDAM aggregated by TCA behaves differently than the detailed network for both active and reactive power flows. The dynamic behavior of the STCA based EDAM is close to the detailed network during and after the fault. Figure 7 shows the voltages at the lower voltage side of the 230 kV/10 kV transformer of the three network models. Similar to the power flow at the boundary bus, the EDAM aggregated by STCA is remarkably close to the voltage measured in the detailed network for the considered events. However, the voltages of the TCA based EDAM show different behavior after the events occur. Especially the frequency jump results in severe deviations of the voltages compared to the detailed model.

D. Validation
The performance of the two approaches, TCA and STCA, is validated by applying the method used in [55] to the EDAM. Originally, the method evaluates CBG simulation models in comparison to their real components. To this end, simulations are conducted with the model and simulation results are compared with measurements. In this work, simulation results of the detailed network represent the measurements of the real component and serve as a benchmark. The simulation results of the EDAM are then compared to those of the detailed network. The validation is based on the values for active and reactive power flows at the boundary bus normalized by the ADN's total active and reactive power demand.
The simulation results are divided into three periods: prefault (A), fault (B), and post-fault (C). A fault is defined as a time period in which the boundary bus voltage is below 0.9 pu. As soon as the voltage rises above this threshold value, the fault period ends. For each period, the following error metrics are evaluated: • mean absolute error MAE: • mean error ME: • maximum error MXE: in which x E (n) is the error between the simulation results of the detailed network model and the EDAM for each data point at time step n (taken every 10 ms) within the total number of N data points per period A, B, and C. For active power, it is defined as where P 0 = 40 MW, and for reactive power as where Q 0 = 10 Mvar. Threshold values for each error type per period are shown in Table II.
In this work, only the short circuit events are considered as faults, since voltage dips for the other two events don't fall below 0.9 pu (Figure 7). Nevertheless, for the phase angle and frequency jump, simulation results are divided in pre-event A (before 0 s) and post-event C (after 0 s) periods to avoid a distorting weighting of the pre-event phase, where deviations are less distinct. The total time range considered for the validation is from −1 s to 5 s. Figure 8 shows the detailed validation results for all five events. Data points are colored according to the aggregation method while their shape depends on the event. The validation is failed if the calculated errors exceed the plotted threshold values listed in Table II. The period B is only applicable for the short circuit faults, since phase angle and frequency jump are only divided into pre-event and post-event periods.
Deviations in the pre-event period A are close to zero for all events in both TCA and STCA based EDAM due to the parameter identification process according to the steady-state power flow. The TCA based EDAM exceeds the threshold value of δ MXE for active power deviation in the post-event period C of the phase angle jump. Also, validation is failed by the TCA based EDAM in all three error metrics δ MAE , δ ME , and δ MXE for reactive power deviation during the one-phase, two-phase, and three-phase short circuit with one exception: δ MAE during the one-phase short circuit. The EDAM aggregated by TCA exceeds the allowed validation limit of reactive power deviation δ MXE in the post-event period of all five events. The frequency jump also leads to reactive power threshold violations for all error parameters in the post-event period. The active and reactive power deviations of the STCA based EDAM are within the threshold values in all events.

E. Complexity Reduction
The main goal of creating EDAM is the reduction of complexity of the detailed network model. Table III shows the reduction of total number of nodes as well as the reduction of simulation time compared to the detailed network model. The severe event of a three-phase short circuit fault is considered for the simulation time comparison. It can clearly be stated that both aggregation methods TCA and STCA reduce the complexity of the detailed network model significantly. As the EDAM aggregated by STCA captures the dynamic behavior of the detailed network model better than the EDAM aggregated by TCA, the slight increase of complexity can be accepted.

IV. DISCUSSION
The simulations conducted in Section III compare the dynamic responses to events of the EDAM derived with the TCA developed by [34] and the EDAM derived with the proposed STCA. It can be seen that the TCA based EDAM fails to adequately reproduce the detailed network's behavior in all five events. However, the STCA based EDAM maps the dynamic behavior of the detailed network model very well, independently of the event.
The results of the conducted validation comply with these observations. Opposed to the EDAM aggregated by TCA, the STCA based EDAM does not exceed any validation thresholds for all five events. The phase angle jump as well as the short circuit faults result in current limitation by the GFMC in the detailed network. Even such highly nonlinear behavior is captured well by the EDAM aggregated by STCA. A significant complexity reduction in terms of simulation time and network size is also achieved by the STCA based EDAM.
The EDAM is derived based on knowledge about the components' parameters in the detailed network, while the remaining unknown parameters of the EDAM are found by analyzing the steady-state power flow and calculating the voltage sensitivities.
However, the analysis conducted in this work is limited with respect to the ADN topology as it assumes an unique connection between ADN and transmission system. ADN with meshed topologies that are connected to the higher voltage grid via multiple transformers are not considered. Nevertheless, applying typical ADN models for stability analysis doesn't necessarily require the consideration of such ADN topologies and is therefore beyond the scope of this work.

V. CONCLUSION AND OUTLOOK
A proper representation of GFMC in EDAM for stability analysis challenges the methods for EDAM derivation. The electrical grid's strength at the GFMC's PCC as well as the electrical distance from the GFMC to the fault location, i.e., the transmission system, are two important factors influencing the dynamic GFMC response. Voltage sensitivities at the GFMC's PCC are identified as suitable parameters for considering these factors in the EDAM. This work proposes the STCA as a novel method to create EDAM containing GFMC on the basis of insight into the detailed network model, i.e., knowledge about technology and control strategy of components as well as load flow data from which voltage sensitivities are derived.
In a STCA based EDAM, GFMC are represented individually dependent on the detailed network's topology. The sensitivities ∂V i ∂P i , ∂V i ∂Q i , ∂ϑ i ∂P i , and ∂ϑ i ∂Q i at the PCC of each GFMC i in the EDAM are matched to those of the respective GFMC's PCC in the detailed network model by parameterizing equivalent impedances accordingly.
Simulations of five different events are conducted on a test network comprising a voltage source as the transmission system and a radial ADN dominated by GFLC and GFMC. Simulation results and validations in this work show the capability of the STCA based EDAM to consider the dynamic behavior of GFMC adequately. Even for severe faults, leading to a nonlinear behavior of GFMC, the performance of STCA based EDAM is within the validation limits.
The STCA as proposed relies on an individual representation of each GFMC in the detailed network model. Hence, an increasing number of GFMC in the detailed network model aggravates the EDAM's complexity. One approach that could solve this challenge is clustering multiple GFMC to one equivalent GFMC dependent on similar voltage sensitivities at the respective PCC [56]. However, the individual GFMC's dynamic behavior in the detailed model cannot be captured well leading to performance drawbacks of the resulting EDAM. This approach should be further investigated and validated in future work. In addition, this work focuses on an open ring network structure for the distribution system. Further work should focus on the performance of the STCA when applied to other network structures such as closed ring networks.

APPENDIX A COMPARISON OF GFMC GENERATION
For a better understanding of the behavior of the GFMC during the events investigated in Section III, the total active and reactive power provision by GFMC in each network model are shown in Figure 9. The observed responses of each GFMC are similar to the overall response of the ADN (Figure 6). While the active power generation of the equivalent GFMC of the EDAM aggregated by TCA deviates somewhat from the generation of the GFMC in the detailed network, significant deviations can be observed with respect to the reactive power injection. For all considered events, the pre-event and post-event values do not match those of the GFMC in the detailed network. In contrast, the equivalent GFMC in the EDAM aggregated by STCA are close to the behavior of the GFMC in the detailed network model. Just a small reactive power offset can be observed for the frequency jump event and the short circuit fault.

APPENDIX B VALIDATION RESULTS WITH DIFFERENT GFMC CONCEPTS
The validation presented in this work focuses on the implementation of droop based GFMC (Section I-B2). To show the validity of the STCA for other GFMC concepts, the same is applied on the detailed network as described in Section III-A, in which the droop based GFMC are substituted by other GFMC concepts with the same set points and PCC. To this end, models of both a Synchronverter [43] and a Virtual Synchronous Machine (VSM) [46] are implemented by utilizing the templates provided in [57]. Figure 10 shows the validation results for STCA based EDAM derived from the detailed network model with Synchronverter as GFMC and VSM as GFMC. Also, the validation results of the S-TCA based EDAM with droop based Fig. 10.
Validation results for STCA based EDAM with three GFMC concepts. GFMC ( Figure 8) are shown. The detailed networks with Synchronverter or VSM as GFMC with the same set points as described in Section III-A were unstable during the one-phase short circuit. Hence, no validation results can be provided for the one-phase short circuit scenario. Beyond that, it can be seen that the threshold values of all error metrics are not exceeded in any time period.