Computing the feasible operating region of active distribution networks: Comparison and validation of random sampling and optimal power ﬂow based methods

The feasible operating region (FOR) indicates the operation points that an active distribution network can achieve at the interconnection point with the transmission grid when operating ﬂexible assets within it; without disturbing the stability of the grid itself. Even though the concept is not new, many novel methods to compute the FOR efﬁciently have been proposed in recent years, resulting in two main schools of thought: random sampling (RS) and optimal power ﬂow (OPF) methods. Both approaches have their merits, yet no wide-ranging analysis regarding scenarios in which each method could be best applied has been done so far. This paper focuses on performing such a comparison; however, capability charts of ﬂexibility providing units and grids are usually modelled as irregular convex polygons, requiring some adaptation of the RS-methods to allow for a proper comparison of the resulting feasible operating region. Correspondingly, new methods to adapt the extrac-tion of random samples from generic capability charts are proposed in the paper. Using models of two radial distribution grids, both OPF- and RS-based methods are compared and validated. Results show that RS methods are adequate for assessing small grids, especially with the proposed improvements, while OPF-based methods excel in larger grids.


FIGURE 1
Schematic representation of the aggregated FOR and FXOR concepts applied to an ADN, based on the feasible and flexible capability charts ( [7, 22]) computation of the FOR, as shown in [13]. To include grid constraints into the calculation, on one hand, a random sampling (RS) approach was proposed in [14], and replicated in [5,7]. The method is simple, yet time consuming due to the large number of iterations required to achieve results of good quality [15]. On the other hand, in [2], an approach to compute the FOR using an optimal power flow (OPF) was proposed. Afterwards, numerous similar approaches have followed in [3,6,13,[15][16][17][18][19][20][21]. All these publications propose different ways to construct and solve the OPF. The main benefits are a greatly reduced computation time and, in most cases, an improved effectivity compared to classical RS approaches. Furthermore, in [1], the Linear Flexibility Aggregation (LFA) method based on a linear OPF was proposed, allowing a reduction of the computation time of the FOR of the larger distribution grids with many FPU.
In [14] and [15], the FOR computation using both RS-and OPF-based approaches were compared. The operation points of the single FPU were constrained within a rectangle in the complex active and reactive power domain, a common simplification of the operational limits of generators in OPF problems. The FPU operation points were randomly selected using uniform probability density functions (PDF) within the quadrilateral power constraints. In both cases, it was described that FOR with larger areas could only be assessed using the OPF method, showing some weaknesses of RS methods. However, it is unclear if the OPF methods are capable of assessing the "real" FOR of a grid, or how good the approximation is. To validate this, a bivariate beta PDF was applied to the RS approach using rectangular FPU capability charts in [23]. Results showed that a beta PDF helps enhancing the area of the assessed FOR compared to using a traditional uniform PDF, resulting in an FOR comparable to the LFA method of [1]. The main result of [23] is that both, OPF and RS methods, could assess the FOR in a similar way, yet the analogy is only valid for rectangular FPU boundaries and for a small radial grid (cf. [24][25][26]).
Additional challenges come when considering that capability charts of FPU can be defined as polygonal shapes. These shapes can be simple, i.e. triangles or squares, or more complex, but can be generally assumed as convex [7,20,22,27]. This provides a better comparison between FOR computation methods, as complex FPU shapes can be integrated as well. Overall, this lets, on one side hand, complementing the validation of OPF-based methods of [23]. On the other hand, improving the performance of RS approaches, allowing their assessment of irregular polygons by adjusting the selection of the random samples. Throughout the paper, different approaches to obtain random samples from convex polygons are described, which are later integrated in the RS method to compute the FOR. The relative merits of using RS and OPF-based methods to compute the FOR of grids with plenty of FPU are analysed using two ADN models. The results should provide guidelines to identify the operating range of both approaches, based on the size of the grid and the number of assessed FPU.

CAPABILITY CHARTS OF FLEXIBILITY PROVIDING UNITS
This section describes diverse non-rectangular capability charts of FPU. In general, an FPU can be defined as a single controllable generator, load or storage system. If a grid section contains several FPU, the available flexibility can be aggregated at the interconnection point of this grid segment to the rest of the system. In such a case a flexibility providing grid (FPG) is defined, whose flexibility potential can be obtained from any FOR calculation method. This can be linked to virtual power plant [7] or microgrid concepts [27].

Capability charts of FPU
In [28], the reactive power capability requirements of different European grid codes are examined. It shows that the expected operational boundaries of FPU can be represented through polygons in the complex power domain. Each FPU is expected to operate within specific PQ ranges, determined by its technical features, as well as the needs and expectations of each grid operator. According to [7], different capability charts are needed for each FPU, in order to compute the FOR and the FXOR. In the case of RES units, the capability chart changes according to the primary energy source (e.g. change in the wind strength or a cloud covering PV panels) or due to explicit control signals [22]. The operation point is still bounded by the technical limits, regardless what causes it to change. After extensive literature review (summarized in Table 1), several models representing different non-rectangular capability charts are proposed in Figure 2. As the inputs for the FOR and FXOR assessments are basically the same in form, i.e. convex polygons, this paper denotes both concepts as the FOR, without losing generality.

Capability charts of power systems
An approach to use capability charts to represent the flexibility provision of FPG and use this information step-wise to aggregate the flexibility provision bottom-up over different voltage levels was suggested in [6, 27. 29]. The resulting FOR are added to the higher voltage levels as FPG boundaries with non-regular convex polygonal shapes. In theory, this approach could be repeated all the way up to the highest voltage level. A graphical  representation of a multi-level aggregation approach is shown in Figure 3. The bottom-up procedure reduces the complexity of the OPF computation at the highest voltage level, as FPG are modelled as simple polygons, e.g. [5,6,27,29]. Those publications showcase the necessity to consider parametrized and non-parametrized types FPU boundaries. However, an insightful analysis of multi-level approaches is not in the scope of this paper.

RANDOM SAMPLE APPROACH TO COMPUTE AN FOR
This section describes a general RS approach to compute the FOR of an ADN, valid for convex polygonal FPU boundaries. Additionally, the selection methods for the independent variables P FPU and Q FPU from generic two-dimensional PDF are described. These random variables represent the variability of the FPU operation points.

FOR computation using random samples
A method to compute the FOR using a RS approach was presented in [14], where the mathematical formulation based on traditional power flow equations in polar coordinates is provided. The flexibility of the FPU is inserted to the power flow calculation by adding the random variables P FPU and Q FPU to the nodal balancing equations, as in Equations (1) Four different PDF represented as heat maps, the colour map shows the probability density: (a) normal, (b) uniform, (c) beta, (d) rademacher and (2). However, not every load, generator or storage system connected to a bus provides flexibility; therefore, a distinction between inflexible load/generation and FPU is necessary. The complex operation points of non-flexible assets N F i connected to bus i are defined as P 0,k + jQ 0,k , ∀k ∈ N F i , while the flexible operation points of FPU F i in bus i are defined as The operation points of the FPU are bounded by a corresponding convex polygon in all cases (cf. Figure 2) [23].
After setting the specific operation points for each FPU, the power flow computation gives the complex voltages of the grid buses, from which the IPF is computed. This sets the basis to determine the flexibility potential of an FPG. The IPF is validated by verifying if all grid constraints are satisfied, e.g. maximal branch loading and voltage magnitude limits. Voltage angles remain unconstrained for this problem. If no grid constraints violations are observed, the IPF is marked as valid, otherwise it becomes non-valid. In an OPF approach, the grid constraints are already included in the optimization problem. The process is repeated N times during the RS method, each iteration using new operation points for each FPU. The FOR is defined as the convex hull surrounding the cloud of points formed by the valid IPFs in the complex power domain (cf. [14]). As is typical in RS approaches, large numbers of iterations are necessary to allow for a proper assessment of the FOR (e.g. ≈10 5 in [14] or ≈10 6 in [23]). A crucial aspect in RS methods is the proper selection of the FPU operation points, as this has a large impact on the quality of the assessed FOR (cf. [23]).

Bivariate PDF to select random samples
In [23], a bivariate beta PDF was proposed for the random selection of FPU set points. This allowed increasing the size of the assessed FOR, as more IPF defined by extreme load and generation states of FPU could be obtained, partially solving an issue raised in [15]. The impact of the bivariate beta PDF on the FOR was extensively analysed. However, the analysis can be extended to other PDF as well, e.g. normal, uniform, beta and Rademacher. A uniform PDF is commonly found in literature to compute the FOR. In contrast, a Rademacher PDF allows focusing the random selection on the vertices of a rectangle, equivalent to the bivariate beta PDF with the parameters , ≪ 1 (cf. [23]) or even the Minkowski sums. Figure 4 shows the differences of the aforementioned PDF. The analysis performed in [23] is extended in this paper (replicating the grid conditions as well) to include the normal and Rademacher PDF. The FOR is computed with 100,000 power flow calculations applying the RS approach, using the four proposed PDF on each FPU. The results with and without consideration of grid constraints are shown in Figure 5.
The Rademacher PDF behaves similarly to the beta PDF, as both allow increasing the size of the FOR, while the normal and uniform PDF would clearly underestimate it. Although slight differences can be observed between the Rademacher and beta PDF, both approaches do help improving the quality of the RS method, proving to be valid solutions to the concerns raised in [15], and supports the reasoning behind [5] and [7]. This demonstration is valid only for rectangular FPU limits, for which a comparison to an OPF method was already provided in [23]. Furthermore, this type of PDF is not trivial to construct for non-regular polygons, requiring the development of additional methods for their production, as is described in Section 4. Figure 5 shows that focusing the random samples on the edges of the FPU helps in improving the FOR assessed using the RS method. This anticipates the course that is followed to extend the RS method to all types of convex polygons.

BIVARIATE PDF FOR NON-RECTANGULAR POLYGONS
In this section, the construction of different PDF that allow assessing the flexibility provision of FPU bounded by nonregular convex polygonal boundaries are presented. Two novel approaches are introduced, one that intends to emulate the beta PDF and the other that emulates the Rademacher PDF from the previous section.

Based on 2D histogram
An approach to construct a bivariate PDF similar to the beta PDF is described next, in order to allow its use with any convex polygon. The objective is to have a PDF with maximal probability density at the edges and minimal density (not zero) at the centre of the polygon, while the exterior of the polygon has a probability equal to zero. The PDF is obtained in the form of a 2D histogram, from which the random operation points of FPU in the complex power domain can be obtained. The proposed construction process of the 2D histogram is inspired in [45], where a polygon  is approximated by a set of overlapped disks D inscribed to . The novelty of [45] is that all disks are centred along the medial axis of . A step-by-step description of the procedure is presented here: 1. Medial axis construction: The medial axis ( ) is computed, which describes the centres of all inscribed disks to  that touch the edges of  tangentially in at least two points (i.e. red lines in Figure 6(a)) [46]. It is composed of two types of segments, the ones touching a vertex of the polygon (limb) and others at the interior of , not touching any vertex (interior).

Polygon approximation with disks:
The polygon  is reconstructed using N inscribed disks  centred along  ( Figure 6(b)). The N disks are distributed evenly within the segments of . 3. Initialize topographic maps: Two different topographic maps are initialized,  int and  limb , which result from the union of many disks  centred among the segments of  ( Figure 6(b)). Initially, both maps have a uniform weight greater than zero. 4. Topographic maps construction: For each map, the topographic levels are charted. The interior part  int is built using sequential erosion operators with disks of constant radius r; then each mask is subtracted from the initial  int , creating the levels observed in Figure 6(c) (top). The levels on  limb are constructed by adding up the area of the disks in the limbs that do not intersect with  int , beginning with the A bin of the 2D histogram is randomly selected using a weighted uniform distribution. The dimensions of  and the discretization of the levels in  play critical roles on the computational burden of the method, thus need to be carefully selected. Figure 7 shows an example of the proposed method, including the medial axis, the resulting topographical map with the ensuing produced bivariate PDF, and the resulting probability density with 100,000 random samples. Some clear similarities to the beta distribution, in Figure 4(c) can be observed.

Based on combination of vertices
It was described in Section 3.2 how the usage of the Rademacher PDF helps increasing the size of the assessed FOR of an FPG. The application of a similar approach to non-rectangular polygons is proposed in this section, where two different methods to define PDF considering just operation points situated at the vertices of the capability chart polygons of FPU are proposed. As the number of FPU in a grid increases, so does the possible combinations of operation points that can be achieved. These methods aim to reduce the search space of the RS approach by reducing the number of possible operation point combinations, as only the vertices of the polygons are considered, instead of the entire area.

Vertices combinations
This approach focusses on generating random combinations of operation points located at the vertices of different FPU polygons. The idea behind this approach is that the selection of operation points of the FPU located on the edges of their capability charts, helps increasing the size of the assessed FOR. This would increase the probability to achieve what would be the "real" shape of the FOR. This method is composed of two steps: 1. Vertices extraction: From the capability chart of an FPU  , the operation points located at the vertices of the convex polygon are extracted and defined as v  . All other operation points are neglected; the ones in the interior and the ones at the edges of  . 2. Random samples selection: For each  , a random operation points from within v  is selected with uniform probability. The IPF is computed from this combination of operation points. The process is repeated N times in total, resulting in N total IPFs (Figure 8; left). The method resembles the previously described Rademacher PDF.

Quadrants partition
One possible outcome of the preceding approach is a combination of diametrically opposed vertices in the capability chart of  . This may lead to IPFs located on the interior of the FOR, not helping to increase the size of the assessed FOR.
To cope with this issue, an adaptation is proposed, where extreme load/generation scenarios for the FPU are imposed, e.g. simultaneously having maximal load and minimal generation, which would increase the probability of assessing an IPF located near the edges of the factual FOR. Many similarities can be drawn between this method and the objective functions of the OPF described i.a. in [15] and [20], which focus the search of the FOR in specific directions in the complex power domain. A procedural description of the process is given here: 1. Quadrants partition: The vertices v  are grouped into four quadrants (Figure 8; right), defined according to the vertices with maximal and minimal active/reactive power values. The four extreme vertices are allocated to both the quadrants they divide. This results in four groups of vertices for each FPU (v  ,I−IV ). A quadrant could contain a single vertex,

NUMERICAL RESULTS
The goal of this section is to evaluate the proposed PDF for the assessment of the FOR, and simultaneously to validate the LFA method. For this purpose, the FOR of two radial distribution grids with different sizes is assessed. The first objective is to verify the impact of different PDF in the RS approach. The second objective is to verify the impact of an increasing number of FPU in the FOR computation with RS. This is relevant if the method needs to be applied in larger grids with considerable amount of FPU. To have a reference for comparing the effectiveness of the PDF, the LFA method is used as baseline. This follows the validation method proposed in [23], where the FOR polygons obtained from both RS and LFA approaches are compared using the similarity concept of Equation (3). There, not only the areas of the polygons are compared, but also their location in the complex power space. The grid topology defines the shape of the FOR; however, analysing this impact is not in the scope of the paper. Such an analysis was already performed in [16] and [47].

12-Bus grid model
The European CIGRE test-bench MV grid of [48] is used to compare all the proposed PDF for the random sampling approach. This section proposes a follow-up to the analysis initiated in [23], which focused solely on the application of the beta PDF. A single-line diagram of the grid model is shown in Figure 9. The LV-grids connected to buses 4-12 are aggregated and their flexibility is characterized as an FPG. For the sake of demonstration, the initial operation point of these grids is defined at the origin of the PQ Cartesian plane.
In the first stage, the general usage of the proposed PDF is evaluated by generating random non-regular convex polygons. For this purpose, the FPG in buses 4-12 are provided with randomly generated capability charts, with 4 to 10 vertices each. All polygons must contain the initial operation point of the FPG (in this cases 0 + j 0 MVA). This way two different scenarios are proposed, the first one where the polygons are limited to ±1 MVA∕MVAr, and a second one where they are restricted to ±1.5 MW∕MVAr. The first scenario allows for a direct comparison with [23], while the second one allows verifying the effect of grid constraints in the FOR computation. An example of the random polygons obtained for both scenarios is shown in Figure 10. Such polygon combinations can generate between   Figure 10. The following PDF are analysed: uniform, 2D-histogram (bivariate), vertices combinations (vertices) and quadrants partition (quadrants).
The resulting FOR obtained from using the aforementioned PDF are shown in Figure 11 and Figure 12, and are compared to the results provided by the LFA method (restricted to 64 FOR boundary points and approximating the maximal branch limits with 32 segments [20]). The polygons of Figure 10 are used as input for the simulations. A noticeable aspect is that the FOR obtained from both LFA and RS approaches correlate extremely well in the shape. The FOR provided by the LFA method contains, in most cases, every boundary point assessed by the RS approach considering all PDF. It is clear that the uniform distribution provides the least amount of information by a large margin, which is in line with what was already observed in [15]  and [23]. The FOR assessed using the uniform PDF is noticeably small, not even able to reach the grid constraints, resulting in the same FOR in both of the scenarios. The bivariate PDF shows some improvements compared to the uniform PDF, but still underperforms based on the expectations raised by the use of the beta PDF in [23].
On the positive side, both PDF focusing on the vertices of the FPU capability charts perform very well, showing the best results among all the tested distributions. Nevertheless, the quadrants PDF clearly outperforms the vertices PDF, showing, in some cases, an almost perfect match to the LFA results. The presence of grid constraints does have an impact in the results, as all the methods tend to properly recognize, in this case, the thermal limit of the line connecting buses 2 and 3 (missing area on the right side of the FOR).
The process is repeated 200 times. During each repetition a new set of random polygons is generated, always considering the defined limits of ±1 MVA∕MVAr and ±1.5 MW∕MVAr. The RS approach is implemented using a linearized load flow calculation method based on [20]. This follows two purposes, first to increase the computation speed of the power flow calculation in the RS approach, and second to introduce an approximation error similar to the LFA method. Using an exact power flow method, like Newton-Raphson, would certainly cause a deviation of the results compared to the LFA method, reducing the effectiveness of Equation (3) as comparator.
The results of this analysis are shown in Figure 13, where the existence of grid constraints is once again critical for the simulations. The boxplots show the absolute max/min, the 25/75 percentile and the average values. The graphic shows similar tendencies to what is observed in Figure 11 and Figure 12. The uniform PDF shows the lowest similarity at all times, while both vertices-oriented PDF show the best overall performance. In the best case, the similarity between the quadrants PDF and the LFA method reaches a maximum of 99.23% in Figure 13(a). Increasing the size of the FPU capability charts causes a dramatical drop from 98.46% to 89.08% in the average similarity of the quadrants PDF, as seen in Figure 13(b), but still providing the best results. This is FIGURE 13 Similarity between RS approach with four different PDF and LFA according to FPG limits (a) ±1 MW∕MVAr, (b) ±1.5 MW∕MVAr FPG limits appealing, as the number of used random samples is rather small, just 10,000.
The bivariate PDF shows a strong improvement compared to the uniform PDF, even though, the focus on the polygon vertices proves to be very effective for this purpose. Additionally, the bivariate PDF shows the largest variance in the test, showing a strong dependence to the shapes of the FPU polygons. Overall, the presence of grid constraints helps to improve the similarity values, compared to the unconstrained scenarios, supporting the findings in [23].
The different polygon combinations that can be achieved from the FPG (cf. Figure 10) can produce a broad range of vertices permutations. A best-case scenario for the FOR computation would be a brute-force approach, where a load flow calculation is performed for each combination of vertices. However, this is only feasible up to a certain number of  Figure 14 shows that this issue has a negligible or no impact (in this small grid model), as the results of all PDF remain mostly invariant to the number of permutations.

111-Bus grid model
In the second case study, the behaviour of the RS approach with a large number of FPG is analysed. For this reason, a large 20 kV distribution grid model with 111 buses was developed, the parametrization is based on a real urban grid. A singleline diagram of the grid topology is shown in Figure 15. From the 111 buses, 95 are modelled as an FPG, each with an initial operation point different than zero. Similar to the first case study, each FPG is provided with a randomly selected polygonal boundary, bounded by ±0.2 MW∕MVAr. The initial operation point of each bus is always included within the polygon. In a few cases, the flexible operation of some of the FPG may cause reverse power flows from the LV grids towards the MV grid. The load and generation parameters of the grid, including initial IPFs, are described in Table 2. All power lines have an impedance of 0.04 + j0.03 p.u.∕ km with an average length of 0.4 km, and can transport up to 10.5 MVA; the transformer is limited to 40 MVA. This case study also uses the LFA approach as basis for comparisons with the RS approach using different PDF. The 95 FPG are sequentially included into the computation based on the buses sequence defined in Figure 15 (beginning with bus 3 and skipping all non-shaded buses). The first iteration considers just a single FPG, then two, and so on until a combination  Figure 16.
As can be expected from Monte-Carlo simulations, increasing the number of random samples does help improve the quality of the results. As is the case in all four defined PDF. With less than 10 FPG connected to the grid, the results are comparable to the ones of the 12-bus system in the previous section. Afterwards, a strong decay in the quality of the assessed FOR is observed for all PDF, especially for the "non-quadrants" methods. The similarity quickly sinks below 30% before the 40 FPG mark. On the other side, the similarity of the quadrants PDF remains above 60% at all times, showing a much slower decay and a muchimproved overall performance, even with just 10,000 random samples.
The improvement effect of increasing the number of random samples tends to fade away as more FPG are considered. Even with an increase in the random samples (within feasible computational times), the quadrants PDF performs much better than the rest of the proposed methods, by a large margin.
The processing time of the RS and the OPF-based methods is compared in Figure 17 (average of 10 repetitions). All simulations are performed using MATLAB R2020a in an AMD Ryzen 7 3700X CPU at 3.6 GHz and 32GB RAM. The OPF in the LFA method is solved using Gurobi. The bivariate PDF shows an enormous deviation in the required processing time compared to all other methods. Its complexity requires plenty of resources, making it unsuitable for the online assessment of the FOR. The processes behind the vertices and quadrants approaches are similar, hence their processing time is similar as well. For this reason, both are shown as one in the graphic. The processing times of the uniform and vertices/quadrants PDF are comparable to the LFA method, as all three increase linearly with the number of FPG. An increase in the number of random samples would cause the processing time of all RS approaches to increase as well. The usage of exact power flow methods instead of linear ones would also increase the computation time.
The results show that typical RS approaches are generally unsuitable to assess the entirety of the FOR of larger grids and large quantities of FPU. On the positive side, the application of new techniques in the selection of the random samples can

CONCLUSIONS
The computation of the FOR of a distribution grid using RS and OPF-based methods was studied in this paper. To allow a proper comparison of both methods, different algorithms to obtain random samples from generic convex polygons were developed, which are integral part of the performed analysis.
As the results show, both methods are capable of assessing the FOR of radial distribution grids of different sizes, as a very good congruency in the results is observed. A critical aspect which differentiates both methods is how they scale-up with the number of assessed FPU/FPG. Increasing the number of FPU/FPG causes the numerical combinations of vertices of the convex polygonal limits to explode, clearly degrading the quality of the assessed FOR in RS methods. The OPF-based methods are clearly the best option when it comes to assess the FOR in grid with a larger number of buses or with plenty of flexible assets. The main conclusion is that OPF-based algorithms are more able to provide a good approximation to the "real" FOR of an ADN, compared to RS-methods, which tend to underestimate it, adding additional insight to the findings in [15] and [23]. The added value of this paper is showing that a proper selection of the samples in RS approaches can help improve their quality dramatically, especially when focusing on the vertices of the FPU capability chart (vertices PDF) and the same quadrant of the polygon (quadrants PDF), instead of the classical uniform or normal PDF. The quadrants approach itself allows improving the similarity score of the FOR in at least 50% compared to all other methods. This certainly creates a good base to FIGURE 17 Comparison of average processing times between LFA and RS methods with different PDF (RS with 10,000 random samples; vertices and quadrants approaches have similar processing times) suggest the use the RS approach even in the case of larger grids. In contrast, just increasing the number of samples shows marginal improvement in the same scenarios, while the trade-off with the computation time can be considerably. This is important, as the main trait of OPF-based methods (i.e. the LFA method is this case) is their reduced computation time.
Another significant aspect is that RS approaches are not necessarily bounde to convex inputs, as they could consider nonconvex FPU/FPG boundaries in the calculation, which is a strong limitation for OPF methods. As to ensure convergence to a global optimum, convex constraints are mandatory. This could be of benefit in some situations, e.g. storage systems with power factor limitations. More complex OPF can always be defined, always with a trade-off with the solving time.
Continuous improvements in computation techniques to obtain the FOR allows for the usage of capability charts to analyse the operation of entire sections of the power system, and not only focus on specific generators. The selection of a suitable method to compute the FOR depends on the characteristics of the grid, the computational resources at hand, the desired quality of the FOR and the existing information about the grid. The resulting FOR depends directly on how the FPU/FPG are modelled. Therefore, the next stage of the research should be to apply these methods in scenarios with non-ideal information from the grid and its operation state, as well as including more detailed information of the individual FPU. Finally, the quality of the resulting FOR should be verified using measurement data, in order to assess the usability of this method in the daily grid operation.