e-Highway 2050: Methodology for network reduction according to critical branches: focus on network partition

Challenge
How to build a zonal transmission system reducing network complexity for large-scale transmission systems while preserving the critical branches that are of special interest for planning methodology?

Background and assumptions
High voltage transmission systems are often too large for the purposes of transmission expansion planning (TEP) studies. Therefore, network reduction methods are necessary to capture the key features of the transmission system into a model that is tractable when performing simulations. This reduced network should result in a transmission expansion as close as possible to the one that would have been obtained for the original nodal system (i.e. with all nodes of the transmission system).

When performing a network reduction, some inter-zonal flows are of particular importance in the perspective of expansion needs and defined as “critical branches”: these critical branches correspond for instance to lines where frequent congestions occur or lines with power flow control devices (HVDC (High-Voltage Direct Current) lines, PSTs (Phase-Shifting Transformers) which require specific modeling since bringing flexibility in system operations.  

The proposed network reduction method is based upon two closely intertwined problems: first assign nodes to zones while preserving the critical branches, second calculate the inter-zonal parameters (capacities and reactances of corridors) that mimic the original transmission system as accurately as possible.

A joint numerical computation of the two problems being unmanageable, a decoupled approach is proposed where the network partition is first calculated and then network equivalent parameters are computed. This network reduction approach is tested on a real case study based on the French and Spanish transmission systems.

1. Network partition according to critical branches
Existing approaches for calculating a zonal equivalent network usually use electrical distance [1] to guide the network partition process.  Thus distances between pairs of nodes that are electrically connected by short, low reactance lines are shorter than distances between nodes that are only connected indirectly through high equivalent impedances. Alternative approaches exist, for instance spectral partitioning approaches that are widely used in image segmentation or K-means clustering algorithms. More sophisticated optimization problem formulations (with the support of MIP) are also found in the literature but lead to tractability issues (i.e. long computational times even for small-sized networks).

The sole use of electrical distance is not fully satisfactory since considering only network topology without any consideration for system operations, localization of generation and demand, capacities of lines. The proposed method, developed and implemented in e-Highway2050, is based on a combination of information contained in electrical and geographical distances, and constraints related to the representation of critical branches in the reduced network model. The associated numerical algorithm is based upon three successive steps:
- identification of critical branches (the simplest partition with an explicit representation of all transmission lines labelled as critical),
- initial partition (a heuristic algorithm builds a partition to be used as the starting point of the clustering process),
- refinement of the initial network partition using a modified version of K-medoids [2].

1.1. Identification of critical branches
Critical branches are important for the operation of the transmission system. Identification criteria include average flow parameters (with regard to the line capacity) complemented by a series of indices characterizing congestion occurrence (operation hours where the line is congested) and congestion severity.

1.2. Initial network partition
Network reduction is based on a composite distance that is calculated as a weighted average of several distances, i.e. geographical distances to favor geographically compact partitions, average Locational Marginal Prices (LMPs) so that nodes with similar prices have a higher probability of being clustered together. Alternatively, Power Transfer Distribution Factors (PTDFs) can be used for critical branches, so that the nodes where a power injection or withdrawal has a similar effect on the flow in critical branches have a higher probability of being clustered together.

In the present case study (see below), a composite distance based upon the electrical distance and the geographical distance has been retained. The highest voltage level in the network is solely taken into account (380-400 kV), so as to avoid distorting the electrical distances. The remaining lower voltage nodes are assigned to the geographically closest partition by geographical distance. An initial partition is then built by a heuristic starting with a single zone to be split as many times as critical branches are identified:  

If both extremities of a critical branch belong to the same zone, this zone is split in two new zones;
the nodes belonging to the split zone are reassigned to the subzone, between the two created, that is closest to these nodes in terms of composite distance.
This algorithm ensures that critical branches are explicitly represented in the partition and takes into account all relevant information for the composite distance calculation. This initial partition is then refined by a clustering algorithm.

1.3. Refining the initial network partition
A modified version of K-medoids (avoiding to assign both end nodes of a critical branch to the same zone) refines the partition. The following steps are executed iteratively [5]:
- a representative node is chosen for each zone (the medoid which presents the lowest average distance to the rest of the nodes within the zone),
- an iterative process updates the partition by assigning each node to the zone whose medoid is closest in terms of composite distance. This step has been modified with respect to classical implementations of K-medoids to guarantee that the end nodes of a critical branch are never assigned to the same zone.

The difference with the better known K-means algorithm is in the selection of the representative node. In the case of K-means the representative node is calculated as the mean of the nodes in the partition, while K-medoids chooses the node with the lowest average distance to the remaining nodes in the partition. Here an alternative criterion is used for the definition of the final representative node of each zone: the “most connected node”, i.e. the node that has the highest connection capacity to the rest of the network, which guarantees that the representatives will be important nodes in the network, such as large power plants or highly connected substations.

2. Calculating Network Parameters
The approach described in reference [8]was implemented, the inter-zonal corridors are defined as the equivalents of the sets of lines connecting pairs of zones in the nodal network model.

2.1 Reactances of inter-zonal corridors
The reactances of the inter-zonal corridors can be calculated by solving a system of linear equations that makes use of the PTDF matrix. Contrarily to other reduction approaches, the proposed technique minimizes errors in inter-zonal flows for multiple scenarios simultaneously. The PTDF matrix of the zonal network can be expressed as a function of the PTDF matrix of the nodal network, the topology of the nodal network and the allocation of nodes to zones. The optimal values of the reactances of the inter-zonal corridors in the zonal network are a function of the zonal PTDF matrix.

2.2 Capacities of inter-zonal corridors
The calculation of capacities of the inter-zonal corridor maximizes transfer of power among zones in the nodal network.

3. Case Study
The case study is based on the French and Spanish very high voltage transmission networks which gather 542 nodes (380-400kV). The target number of zones was 50 in order to obtain a one-order-of-magnitude reduction in terms of number of nodes.

The indices described above for critical branches were calculated across 10 different scenarios with one time horizon, where the hourly operation is simulated for a year. The different criteria proposed for the identification of critical branches lead to slightly different solutions. However, even though the specific critical branches identified according to different criteria were not identical, the same relevant corridors were sketched.

The preliminary study resorted to the top 100 most relevant critical branches selected by the average flow criterion, supplemented by 17 other elements, namely PSTs, which were added to the list of critical branches.
The initial partition resulted in 24 zones which were then refined into 50 zones using a composite distance (50% electrical distance and 50% geographical distances). The resulting partition (Figure 1) using the composite distance is more compact geographically than if only the electrical distance is used.


Figure 1: Network reduction for the case study using a composite distance


Figure 2: Representation of the reduced network

The final result of the reduction is shown in Figure 2. This zonal network is now suitable for  network expansion planning approaches.

4. Conclusions
Given the computational complexity of long-term planning of large-scale transmission systems, network reduction is necessary to perform optimal expansion planning.

The proposed network partition method is based on critical branches, i.e. particularly relevant for TEP purposes because of frequent congestions or of particular interest for the operation of the system. Critical branches are identified based on their average flow, the proportion of hours during which a line is congested, or the economic impact of congestions. A heuristic algorithm provides an initial partition that keeps the extremities of critical branches in different zones. Then, this partition is later refined by applying a modified version of K-medoids. This algorithm clusters nodes based on a combination of electrical and geographical distances.

The proposed network reduction technique has been applied to a real case study based on the Spanish and French systems. The resulting zonal system retains the explicit definition of the critical branches and is now amenable to optimization.

References
[1]    Haixia Wang, Rao Liu, Weidong Li and Caihong Zhao, "Power flow tracing with consideration of the electrical distance," in Power and Energy Engineering Conference, 2009. APPEEC 2009. Asia-Pacific, 2009, pp. 1-4.
[2]    C. Hamon, E. Shayesteh, M. Amelin and L. Söder, "Two partitioning methods for multi-area studies in large power systems," International Transactions on Electrical Energy Systems, 2014.
[3]    E. Cotilla-Sanchez, P. D. H. Hines, C. Barrows, S. Blumsack and M. Patel, "Multi-Attribute Partitioning of Power Networks Based on Electrical Distance," Power Systems, IEEE Transactions on, vol. 28, pp. 4979-4987, 2013.
[4]     S. Blumsack, P. Hines, M. Patel, C. Barrows and E. Cotilla Sanchez, "Defining power network zones from measures of electrical distance," in Power & Energy Society General Meeting, 2009. PES '09. IEEE, 2009, pp. 1-8.
[5]    J. A. Hartigan and M. A. Wong, "Algorithm AS 136: A K-means clustering algorithm," Applied Statistics, vol. 28, pp. 100-108, 1979.
[6]    A. Papaemmanouil and G. Andersson, On the reduction of large power system models for power market simulations, Power Systems Computation
[7]    Conference (PSCC), Stockholm, Sweden, 2011. http://www.pscccentral.org/uploads/tx_ethpublications/fp156.pdf 
[8]    Di Shi and D. J. Tylavsky, "An improved bus aggregation technique for generating network equivalents," in Power and Energy Society General. Meeting, 2012 IEEE, 2012, pp. 1-8.
[9]    S. Lumbreras, A. Ramos, L. Olmos, F. Echavarren, F. Banez-Chicharro, M. Rivier ; P. Panciatici, J. Maeght, C. Pache “Network Partition Based on Critical Branches for Large-Scale Transmission Expansion Planning“, Powertech, 2015