Benchmark Low Voltage Distribution Networks Based on Cluster Analysis of Actual Grid Properties J. Dickert, M. Domagk, P. Schegner Institute of Electrical Power Systems and High Voltage Engineering TU Dresden Dresden, Germany
[email protected] Abstract—Distribution systems are supplying many customers with electricity requiring a high number of equipment and lines. It is tedious to analyze all low-voltage (LV) networks of a distribution system operator (DSO). Because of the large number of LV-networks within the system benchmark networks are required. For bulk analysis the networks have to represent a large number of networks within the system. For extreme value analysis the most critical networks have to be identified and examined. Benchmark networks representing typical LVnetworks for bulk analysis can be derived from technical data of these networks using multivariate methods. This paper describes the data allocation and application of cluster analysis for this purpose. The presented benchmark networks are based on German conditions. The networks are also reviewed under the aspect of feasibility. The benchmark networks are characterized by their supply obligation as well as the network data and properties are given for benchmark analysis. The paper also includes the diverse experience in data allocation, analysis of the data and gives assistants for prospective data allocation combined with cluster analysis of LV-networks. With the information given in the paper also benchmark networks adapted to a variety of other applications can be derived. Index Terms--clustering methods, data handling, power system planning, multivariate methods
GLOSSARY – FROM ELECTROPEDIA [1] radial system: a system or part of a system consisting of single feeders supplied from a single source of supply (601-02-15) tree'd system: a modified radial system to which spurs have been added (601-02-16) single feeder (radial feeder): an electric line supplied from one end only (601-02-09) tapped line: a main line to which branch lines are connected (601-02-11) branch line (spur): an electric line connected to a main line at a point on its route (601-02-10) line connection (supply service): a branch line from the distribution system to supply a consumer's installation (601-02-12) line section: portion of an electric line bounded by two points which are either terminations of the line or line taps (601-02-30) delivery point: interface point between an electric power system and a user of electric energy (601-02-33) metering point (point of measurement): point in an electric power system, where flow of energy and, when applicable, the flow of electric power is metered (617-04-06) embedded generation (distributed generation, dispersed generation): generation of electric energy by multiple sources which are connected to the power distribution system (617-04-09)
I.
INTRODUCTION
Distribution systems are supplying customers with electricity area-wide; where else the transmission system can be described as backbone for the electricity supply. Distribution systems can be subdivided into medium-voltage (MV) networks and LV-networks. MV-networks are also called primary distribution systems with nominal voltages of 10 kV or 20 kV and LV-networks are secondary distribution systems with the nominal voltage of 400 V. Distribution systems are much larger than transmission systems, especially looking at the number of assets. The line lengths for cables and overhead-lines in Germany [2] sum up to: x x x
1.000.000 km of LV-networks (Vn d 1 kV) 500.000 km of MV-networks ( 1 < Vn d 60 kV) 100.000 km of HV-networks (60 kV < Vn)
in respect to the nominal voltage Vn. These numbers illustrate that more than 90 % of the lines are within the distribution system, which combines the LV- and MV-networks. Also more than 98 % of the installed transformers in the grid are connecting MV- with LV-networks. General studies concerning distribution systems can only be performed on some exemplary networks. The proper selection of these networks is crucial for the validity of the results of the study. This shows the need for benchmark networks representing the important influencing factors. The objective of this paper is therefore the elaboration of such networks from network data. Hence, it will give information about the necessary data collection and the important parameters. The compiled representative feeders are supposed to present the LVnetwork. The power system test cases from the University of Washington [3] are used for general examinations on transmission systems. For distribution systems test cases are only established for North America [4], [5]. For analysis on microgrids a detailed study case network is presented in [6]. This includes only one LV-feeder. For MV-networks [7] and LV-networks [8] some considerations have been performed on benchmark models by the CIGRE Study Committee C6. The LV-network benchmark model focuses also on North American distribution grids and includes one residential, one industrial as well as one commercial sub-network.
building footprint line connection distribution transformer
street with LV-line street without LV-Line
Figure 1. Example for a township including streets and buidlings
II.
CHARACTERIZATION OF LV-NETWORKS
A. General The main objective of LV-networks is the connection of the customer to the distribution grid. LV-networks reach from the terminals of distribution transformers to the delivery points of the customers. Distribution transformers have a preferred rated power between 100 kVA up to 630 kVA. Customers supplied by LV-networks are: x x x
residential customers (households), small businesses, shops and institutions (e.g. schools, hospitals).
B. Geographical Area The utilities are limited in the alignment of the grid by laying lines to the course of the streets. Overhead-lines may cut a corner, but still most customers require a line connection (supply service). Fig. 1 illustrates a township with streets and buildings. The placement of the distribution transformer is dependent on the course of the MV-network but should be at the load center. Several feeders are normally used to connect the distribution transformer with the customers. In LVnetworks, the feeders are usually single feeders (radial feeders) and are supplied from one end only. There are exceptions in megacities where the LV-networks are constructed as meshed networks. This type of networks is not in the focus of this paper.
distribution transformer station
delivery point
Figure 2. Schematic diagram of a LV-network
C. Tapped line Within a feeder there is always one tapped line. The lines coming of the tapped line are so-called branch lines. The identification of the tapped line may be done under different assumptions requiring diverse information from the network. To find the tapped line each line of one feeder has to be reviewed. Meaningful criteria for the identification of the tapped line can be x x x
line length l, impedance Z or expected voltage drop 'V.
In terms of length l, the tapped line has the apparent maximal line length lmax between the distribution transformer and the furthest delivery point of all lines. The impedance Z can be depicted as electric distance of the line. In this case the tapped line has the highest impedance Zmax of all lines. The expected voltage drop 'V is calculated with the allocated load current IL and the according impedance ZLS of the respective line segment. It is also possible to perform the calculation with the absolute values of current and impedance. 'V ZLSn ZLS(n-1)
ILn
IL(n-1)
ZLSi IL(k+1)
ZLS1 IL(k)
IL2
IL1
Figure 3. Equivalent circuit for calculation of the voltage drop
Without branches the system is called a radial system and with branching it is a tree'd system. Nevertheless, one distribution transformer may supply some feeders with branching and some without branching. This can be exemplarily seen in Fig. 2. This schematic illustrates the LVnetwork of the township shown in Fig. 1. The schematic diagram does not include any geographical information about the customers, but it enables the network planner to get a better overview of the grid.
The line with the highest expected voltage drop 'Vmax is the tapped line. The tapped line has therefore the lowest supply voltage.
The analysis of LV-networks should be performed on base of the data of each feeder. Data allocation for distribution transformer stations is not sufficient since information like lengths even out. This is due to the fact that one distribution transformer stations supplies different customer types.
It is also possible that the current feed-in by distributed generation is dominant. In this case, not the load current has to be taken to calculate the expected voltage drop, but the feed-in current IG by the embedded generation. The result is a voltage rise rather than a voltage drop.
Fig. 3 shows the equivalent circuit for calculating the expected voltage drop 'V with (1). 'V
n
§
i
¦ ¨© ¦ I i 1
k 1
k
· ¸ Z Li ¹
(1)
D. Branching The occurrence of branching has to be recognized for the characterization of LV-feeders. Three branching types can favorably be distinguished. Fig. 4 shows the different branching types. Proper single feeders have no branching and consist of only one tapped line. So-called simple branching allows no further branching of each branch line. Nevertheless, several branch lines may be connected to the tapped line. Feeders with multiple branching have also branching within the branch lines. The approach to consider branching is explained in the section data acquisition. E. Load and Generation The customer load and feed-in of distributed generation is also of interest for benchmark LV-networks. They depend on the customer types [9] (partly-electric, fully-electric, allelectric, etc.) as well as on the regional conditions for the potential of distributed generation. The further development of load and generation changes is also depending on new technologies (heat pumps, small scale embedded generators, electric cars). With the increase of electrification the power flow within the distribution grids increases. F. Electric Parameters The electric parameters for standard cables and standard overhead-lines are given in Table I. The data includes the rated current Ir, the DC-resistance R20 at 20°C and the inductance X. The capacitance of LV-lines can be neglected for load flow calculations. TABLE I.
PARAMETERS FOR STANDARD CABLES Parameters Rc20 : km
Ir A
OHL
Cable
a.
NAYY 4x35 mm² NAYY 4x120 mm² b. NAYY 4x150 mm² NAYY 4x240 mm² a. NFA2X 4x35 mm² b. NFA2X 4x70 mm² a. Al 4x35 mm² (bare) Al 4x70 mm² (bare) a. for line connections
III.
123 245 275 364 132 205 145 290
0,868 0,253 0,206 0,125 0,868 0,443 0,830 0,443
Xc : km 0.08 0.08 0.08 0.08 0.07 0.07 0.34 0.32
b. preferred lines for feeders
Reflecting the study, the following information would have made the results more realistic: x average length of line connections, x number of metering points, x branching type, x number of branches and x distance to first branch line. The description of branches with parameters is problematic. But the information about the number of branches and the location of the first lateral allows sufficient attention to the branches. As this data was not available for the conducted study a schematic approach is chosen and explained in section V.
IV.
DATA ANALYSIS
The data set of the feeders available for the conducted data analysis consists of 6 numerical variables (ltotal, ltotal cable, ltotal OHL, nDP, nDP/ltotal and SrT). Hence, each feeder can be represented in 6 dimensions with each dimension represented by one of the variables. The data analysis consists of 2 steps. In the first step the principal component analysis is used to reduce the number of 6 dimensions to 2 dimensions. This enables an overview of the data set and allows the identification of variables which do not contribute with significant information for grouping. The reduced data set is clustered in the second step to consolidate the feeders into groups based on their properties. Feeders in one group should be similar whereas feeders of different groups should be dissimilar. A. Principal Component Analysis The principal component analysis (PCA) is a transformation which is used to get an overview of the structure of the available data in a reduced number of dimensions which are called principal components PC. The principal components are uncorrelated linear combinations of the original 6 variables. The information criterion for the PCA is the total variance of the data set. Original variables with a high variance are weighted higher than variables with a low variance within the PCs [10]. The importance of the components is based on the associated variance part of the
DATA ACQUISITION
To compile benchmark networks actual data from real LVnetworks has to be analyzed. Nowadays utilities have the possibility to provide the required data using the information stored in geographic information systems (GIS). However it is important to know which data is required for the compilation of benchmark networks. The data available for the conducted study for each feeder is: x x x x x
For the data analysis, the rated voltage Vr is not included. However, the ratio nDP/ltotal is also used for the data analysis.
total length ltotal, cable length ltotal cable, overhead-line length ltotal OHL, number nDP of delivery points as well as rated voltage Vr and rated power SrT of the distribution transformer.
MV grid
a
no branching
a
simple branching
a
multiple branching
b b b b b
delivery point distribution transformer
a b
tapped line branch line
Figure 4. Classification of branching
The k-means clustering algorithm is used for the actual clustering process which partitions n feeders into k groups. The iterative refinement process starts with the random selection of k feeders to represent the group centers. The remaining feeders are assigned to their nearest center based on their Euclidean distance between them. In the next iteration a new center is calculated for each group and every feeder is reassigned to their new nearest center [11]. This process is repeated for 10 iterations. The R-squared RS is used to validate the different cluster solutions by calculating the ratio between the sum of squares between the groups SSb and the sum of squares for the whole data set SSt as seen in (3). RS Figure 5. Transformation of the data into 2 principal components representing 90% of the total variance (feeders: black points; original variables: red lines)
total variance in their direction. PC1 has the highest variance associated in their direction, PC2 the second highest variance etc. The data set is transformed into their first two principal components PC1 and PC2 (see Fig. 5). PC1 includes 66 % and PC2 24 % of the total variance. Both principal components PC1 and PC2 include 90 % of the total variance and are a good representation of the 6-dimensional data set. Each feeder is shown as black point in Fig. 5. The original variables are displayed as red lines. The individual lengths of the lines indicate their representation within the PCs. Hence, ltotal, ltotal cable, ltotal OHL, nDP are the major contributors to the PCs. The rated power of the distribution transformer SrT and the number of delivery points per total line length nDP/ltotal are very short. Therefore, they are pointing into the 4 remaining dimensions which represent only 10 % of the total variance for the data set. Therefore they can be neglected for further analysis.
SS b SS t
(3)
The values for RS are between 0 and 1. RS near 1 indicates significant differences, whereas 0 shows no differences, between groups [12]. The number of groups has been increased from 2 to 9. For each number of groups the k-means algorithm has been applied 100 times to reduce the effect of the randomness for the initial centers on the clustering solution. The best result has been stored for each number of groups with the associated RS value. A number of 6 groups has a RS value of 0.76. As the gradient of RS with larger group numbers is very small, further increase in number of groups will only have a marginal improvement of the clustering result. Fig. 6 visualizes the clustering for 6 groups using the multidimensional scaling and the names for the groups.
B. Cluster Analysis Cluster analysis is used to gather the feeders into groups based on their properties. A measure of distance is mandatory for the clustering of the feeders. The Euclidean distance d is suitable for this application and is defined as follows: v
d
¦ (a
i
bi ) 2
(2)
i 1
The Euclidean distance between 2 feeders a and b is the square root for the sum of their squared differences for each of the v variables. In this case the remaining 4 variables after the PCA (ltotal, ltotal cable, ltotal OHL, nDP) are used to calculate the Euclidean distance between each of the feeders. To guarantee that each of the variables is treated equally, their range is scaled between 0 and 1 before calculating the Euclidean distances. Differences of variables with high orders of magnitude (e.g. total length) are weighted stronger than differences of variables with smaller orders of magnitudes (e.g. delivery points) due to the sum of the squared differences.
Figure 6. MDS with clustering result
V.
BENCHMARK FEEDERS
A. Classification With the results of the cluster analysis it is shown that 6 groups represent the feeders adequately. The classification is shown in Fig. 7 with descriptive names. For short-range feeders overhead-lines compose a negligible portion within the evaluated data. Longer feeders may contain overheadlines. It strongly depends on the rules of action of the DSO. For the benchmark feeders, overhead-lines are aligned at the end of each feeder.
increase of branches .
decrease of distance between DP .
increase number of DP .
secondary system
increase of total feeder length .
B0: ltotal = 100m lB0 = 100m nDP B0 = 10 B0 B1
100m
B1: ltotal = 100m lB1 = 70m nDP B1 = 7 40m
30m 30m
B2
B2: ltotal = 100m lB2 = 60m nDP B2 = 6 20m
20m
20m 10m
30m
B3: ltotal = 100m lB3 = 40m nDP B3 = 4 Figure 7. Classification of the feeders with sub-division
Fig. 7 also shows some specifics of the feeder. As feeders get longer, the number of delivery points nDP increases but the distance between delivery points decreases. As of branching it is assumed that with longer feeders it is more likely that branching occurs. Nevertheless it is not possible to classify into urban, suburban or rural feeders. There is a tendency that feeders containing overhead-lines are located in rural areas. But there are also many short-range feeders to be found there for the connection of e.g. farms. B. Branching Schematization No branching and simple branching is used for the compilation of the benchmark networks. About 90 % of the feeders have this characteristic [13]. With this awareness a reasonable approach to model branching is ideally shown in Fig. 8 and considering the line section distances dDP between delivery points in Fig. 9.
B3
10m 10m 10m 10m 10m 20m
dDP = 10m delivery point (DP) nDP Bi number of delivery points of tapped line
30m Figure 9. Branching considering the line section distances
C. Parameters for Benchmark Feeders The description of the benchmark feeders requires only a few parameters: x the distance between delivery points dDP, x the number nDP Bi of delivery points of the tapped line, x with respect to the branching i. The implemented branching for the different feeder types is given in Table II. TABLE II.
BRANCHING OF FEEDER TYPES
Feeder Type short-range – single customer short-range – multiple customer middle-range – multiple customer long-range – multiple customer TABLE III.
ltotal 3 B2: lB2
3 ltotal 6 B3: lB3
4 ltotal 10 (4)
The lengths of the tapped and branch lines for Fig. 8 are depending on the distance between the delivery points dDP. For simplification branching is only allowed at delivery points. The tapped line has to be the longest line or in other words, branch lines are shorter than or as long as the tapped line. Therefore the tapped line gets the most delivery points. This is the reason why the tapped feeder with 2 branches is ideally 10 m (Fig. 8) shorter than the feeder considering the length of line sections as seen in Fig. 9.
mid
B1: lB1
long
For all feeders in the figures, the overall length dtotal is 100 m but the length of the tapped lines dBi as well as the number of delivery points of the tapped lines nDP Bi reduces as the number i of branches increases. The lengths of the tapped lines for Fig. 8 are calculated by dividing the total length of the line ltotal as follows:
short
Figure 8. Ideal branching
dDP nDP B0 1 60 1 120 2 80 3 80 6 50 10 40 15 40 20 35 25 30 30 30 40 30 50 30
B0 ¥ ¥ ¥
Branching B1 B2 B3 ¥ ¥
¥ ¥
¥
PARAMETERS FOR BENCHMARK FEEDERS – CABLE lB0 nDP B1 lB1 nDP B2 lB2 60 good case 120 average case 160 worse case 240 2 160 good case 300 4 200 average case 400 7 280 worse case 600 10 400 9 360 700 14 490 11 385 750 17 510 14 420 900 good case 15 450 1200 average case 21 630 1500 worse case 26 780 all lengths and distances in m
nDP B3
lB3
good case average case worse case 12 360 16 480 20 600
The parameters for the feeders are specified in Table III and Table IV with 3 feeders for each type representing each a good case, average case and worse case. Feeders declared as good case are shorter and have less delivery points per feeder whereas worse case feeders are longer and have more delivery
points. For long-range feeders the parameters for no branching are also included for comparison reason, because ltotal = lB0. Nevertheless, such feeders are not found in practice. Fig. 10 illustrates exemplarily the composition of the middle-range good case benchmark feeder regarding branching. The allocation of the number of DP per branch is shown in Fig. 11. B0 B1
B0: ltotal = 600m lB0 = 600m nDP B0 = 15 600m
B1: ltotal = 600m lB1 = 400m nDP B1 = 10 200m
200m 200m
B2
B2: ltotal = 600m lB2 = 360m nDP B2 = 9 120m
120m
120m 80m 160m
mid-range good case dDP = 40m
Figure 10. Example of benchmark LV-feeders for mid-range cable
The feeders containing overhead-lines have longer distances between delivery points dDP and are also longer compared to their counterparts with cable only. This is reasonable since overhead-lines are likely to be found in rural areas with larger properties. The overhead-line ratio is about 60 % of the total length of the feeder. The overhead-line sections are placed at the end of the feeders in a way, that the tapped line has the most portions of overhead-lines.
Long
mid
TABLE IV.
PARAMETERS FOR BENCHMARK FEEDERS – C&OHL lB0 nDP B1 lB1 nDP B2 lB2 500 7 350 6 300 585 9 405 8 360 640 11 440 9 360 800 good case 11 440 1200 average case 15 600 1600 worse case 21 840 all lengths and distances in m
dDP nDP B0 10 50 13 45 16 40 20 40 30 40 40 40
nDP B3 lB3 good case average case worse case 8 320 12 480 16 640
The allocation of the number of delivery points to branches is shown in Fig. 11. The number of delivery points is calculated depending on the number of branches. The ceiling function “ceil” (round up) is used to calculated the delivery points per line section. This results in to many delivery points and requires a reduction procedure. For 3 branches it starts by reducing 1 DP at branch 1, 2 DPs at branch 2 and 3 DPs at branch 3 before reducing 1 DP in the tapped line. If this is not enough, the process starts again at branch 1 by reducing 1 DP.
nLS
nLS
§n · ceil ¨ DP ¸ © 3 ¹
nLS
nLS
nLS
nLS
§n · ceil ¨ DP ¸ © 6 ¹
nLS
nLS
2nLS
nLS
nLS
nLS nLS
§n · ceil ¨ DP ¸ © 10 ¹
The length of the line connections is set to 15 m. Line connections for single customers are included in the tapped line. VI.
CONCLUSION
In this paper LV-networks are characterized. The data required to compile benchmark networks is specified and resulting LV-networks are presented. The classification of the feeders is carried out with the k-means clustering algorithm. The resulting classifications are related to practice and allow analysis on validated benchmark networks. A variety of data is given for short-, medium- and long-range feeders regarding good case, average case and worse case scenarios. Combined with the given electric parameters it is possible to arrange benchmark networks adapted to the scope of the requested research. It is also shown how benchmark networks can be derived from technical data, which can be acquired by using multivariate methods. It is pointed out which data has to be available for such analysis. Branching is one of the biggest challenges concerning LV-feeders. An easy to use approach is presented which can be used if data is not available. REFERENCES [1]
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nLS
nLS
3nLS 2nLS
nLS
Figure 11. Procedure for the allocation of delivery points to branches
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