Network partition based on critical branches for large ... - IEEE Xplore

Networl( Partition Based on Critical Branches for Large-Scale Transmission Expansion Planning

Sara Lumbreras Andres Ramos Luis Olmos Francisco Echavarren Fernando Banez-Chicharro Michel Rivier Institute for Research in Technology Universidad Pontificia Comillas Madrid,Spain [email protected]

Patrick Panciatici Jean Maeght Camille Pache RTE R&D,Versailles,France

Abstract-The size and complexity of large power systems, such as the European one, often make them unmanageable for the purposes

of

Therefore,

Transmission

network

Expansion

reduction

methods

Planning are

(TEP).

necessary

to

condense their key features into a workable model, which should approximate the behavior of the nodal system as accurately as

possible. We define critical branches as the transmission lines that are particularly relevant for TEP purposes because of, for instance, frequent congestions or a special nature such as the cases of HVDC (High-Voltage Direct Current) transmission lines

or

PSTs

(Phase-Shifting

Transformers).

It

would

be

desirable to preserve these critical branches while simplifying the remaining grid. We propose a heuristic algorithm that creates an initial partition that is later refined by clustering nodes based on a composite distance measure. The proposed technique has been applied to a real case study based on the French and Spanish systems.

Index

Terms-

Circuit

Optimization,

Clustering

Methods,

Power Transmission.

I.

INTRODUCTION

High voltage transmission systems are often too large for the purposes of transmission expansion planning (TEP). Therefore, network reduction methods are necessary to condense the key features of the system into a model that is sufficiently small to be manageable. This reduced network should approximate the behavior that drives the expansion of the full system as accurately as possible, so that the transmission expansion plan obtained for the zonal network is as close as possible to the one that would have resulted from the original nodal system. This means that, to the extent possible, the inter-zonal flows of the zonal network should approximate those in the nodal system. Some of these inter­ zonal flows are more important than others from the perspective of expansion needs. We define critical branches as the transmission lines that present a special interest for the

This work is part of the E-Highway 2050 project supported by the EU Seventh Framework Programme (www.e-highway2050.eu).

purposes of TEP. Possible reasons for this special interest are, for instance: Frequent congestions. If a line is congested, it will impose operation constraints that would be ignored if the line was internal to a zone. That is particularly important given that congestion is one of the main drivers of transmission expansion decisions. Some power flow control devices, such as HVDC (High­ Voltage Direct Current) lines or PSTs (Phase-Shifting Transformers). These elements should be modeled individually in order to capture the flexibility they bring to the operation of the system. Work Package 8 of the e-Highway2050 project, as explained below, proposes that the network reduction process should preserve the explicit definition of these critical branches while simpliJYing other parts of the grid. This simplification comprises two subproblems: First, network partition assigns nodes to zones. According to our approach, this partition should preserve all critical branches. That is, the end nodes of critical branches should belong to different zones. Then, network reduction calculates the zonal network parameters (namely, the capacities and reactances of inter-zonal corridors) that approximate the characteristics of the original system as accurately as possible. Ideally, these two subproblems should be performed simultaneously to find the partition that results in the best possible reduced system. However, the unmanageability of this approach leads us to a decoupled approach that finds a network partition first and subsequently calculates its equivalent network parameters. This paper proposes an approach that finds a zonal system efficiently for large-scale transmission networks while preserving the identified critical branches. A real case study based on the French and Spanish systems demonstrates the applicability of the developed technique. This paper is structured as follows. First, section II describes the need for network reduction. Section III reviews

the main approaches to network partition in the literature. Then, section IV presents the proposed approach. Section V gives some guidelines on the calculation of network parameters. Section VI reports the case study results. Finally, section VII extracts conclusions. II.

LARGE-SCALE TEP.

THE NEED FOR NETWORK

REDUCTION.

The structure of the transmission network imposes constraints to the long-distance power flows that can have a deep impact on the operation of the system. Consequently, TEP has been extensively studied in both academic and practical contexts [1]. The complexity of optimal TEP means that it is usually performed for areas no larger than a country or province, for time horizons that stay confined in the short and medium term,and for reduced sets of uncertain scenarios. However, the current trends in power systems make it necessary to attempt solving large-scale,long-term versions of the TEP problem for a wide set of uncertain scenarios. The deregulation of the power sector means that Generation Expansion Planning (OEP) is not performed according to a central plan but undertaken by private companies. This turns out to be a challenge given that the lead times for generation plants are much shorter than for transmission lines (around 3 years compared to often more than 10 years). Therefore, TEP must anticipate GEP and prepare for different generation scenarios. In addition, large amounts of renewable generation are expected in Europe for the coming decades. A large part of these will present themselves as large,coordinated,projects: Desertec is a German initiative that aims at installing over 20 OW of renewable power in the Sahara desert and its surroundings. It plans to export part of the generated energy to Europe, partly through newly-built transmission lines [2]. MedGrid and MedRing are initiatives that plan to install over 20 GW of renewable power around the Mediterranean and support power exchanges in the region [3,4]. Projects such as the Offshore Grid Initiative study the potential of offshore wind in Europe, which could also make use of the necessary offshore networks to support power exchanges among the neighboring countries [5]. Many of these projects extend across borders, and overall they will create reinforcement needs for the European transmission network that should be tackled in a coordinated way. In the European Union, the Ten-Year Network Development Plan (TYNDP) currently results from a bottom­ up approach that combines national transmission plans into a long-term European expansion strategy where Projects of Common Interest for the European Union are identified and assessed [6]. This approach combines national plans rather than planning the whole European region in an integrated manner. On the contrary, project e-Highway2050 has the objective of developing a methodology for the integrated TEP of the European transmission network up to 2050, ensuring reliability and the integration of national and regional markets and RES generation into them. The project will have two results: the methodology itself and an expansion plan for

electricity highways taking into account a set of different future power system scenarios [7]. Within this project, Work Package 8 (WP8) aims at formulating TEP as a centralized optimization problem, taking into account several OEP scenarios and the uncertainties inherent to renewable generation,hydro inflows and demand. However, the optimal planning of an area as large as the European system and the uncertainties involved make it impossible to solve the whole problem directly. The strategy developed in WP8 relies on reducing the network to a simpler, equivalent model that is amenable to optimization. The main steps of the proposed method are detailed in Figure l.

I

STEP I - ADEQUACY WTTHOUT GRID

I

STEP 2 - DETECTION OF SYSTEM OVERLOADS

I

STEP 3 - NETWORK REDUCTION ACCORDING

I

STEP 4 - OPTIMAL GRID EXPANSION AT ZONAL LEVEL FROM TODAY TO 2050

I

I

STEP 5 - GRID EXPANSION AT NODAL LEVEL

..

I

•

I

I

.. •

TO CRITICAL BRANCHES

..

STEP 6 - ROBUSTNESS OF THE PROPOSED GRID ARCHITECTURES

I I

Yearly MonteCarlo simulations

I

Figure 1.Methology proposed in eHighway2050, WP8.

First, copperplate and transmission-constrained simulations of the European power system allow identifYing the most important congestions in the network for a wide set of possible evolutions of the system. These congested lines, together with Flow Controlling Devices (FCD) make up the critical branches, which are used as the key element in the reduction. Once this reduction based on critical branches has been performed, the result is a zonal system that approximates the behavior of the complete, original nodal system. The zonal system is small enough to be tractable, but keeps a description of all the relevant congestion of the network thanks to its representation of critical branches. A modular expansion plan up to 2050 is calculated by optimizing the zonal system. This optimization is calculated for a subset of scenarios and snapshots that has previously been selected using clustering techniques. The zonal plan is subsequently translated into equivalent nodal plans for the closest time horizons. Finally, the proposed plans are tested for robustness. The whole approach relies on reducing the stochastic and temporal complexity via snapshot selection and the complexity of the network via the calculation of an equivalent zonal network. This article describes the proposed network reduction method and applies it to a real case study based on the French and Spanish systems.

III.

EXISTING ApPROACHES

Calculating a zonal equivalent network implies two differentiated steps. First, the nodes are assigned to zones by network partition. Then, the parameters that describe the corridors linking zones are calculated. Most of the references that deal with network partition use electrical distance to guide the process. The electrical distance between a pair of nodes in a network is defined as the equivalent impedance between them, i.e., the voltage drop between the nodes when a current of 1 A is transported through the network from one of the nodes to the other. This equivalent impedance is computed using the elements of the inverse of the admittance matrix,i.e., the Zbus matrix [8]. The electrical distance Dij between nodes i and} is computed as follows: D

B

=

Zbus + Zbus - 2 Zbus U

ll

B

(1)

where the elements of the matrix Zbus correspond to: Zbus Ybusi) YbUSii

=

Ybus·1

=

=

(Rl) +.jX,) r L:(Rij+jXiJ+L:jBij,i -

(2) (3) (4)

where R, X, G and B represent the resistance, reactance, conductance and susceptance of the corresponding lines, respectively.

,,$ �

Figure 2. Calculation of the electrical distance between nodes i and}. The distance ref1ects the voltage drop between the nodes when a current of one unit is injected in one of the nodes and withdrawn in the other

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. It is important to note that it is necessary to include the conductances and susceptances of all system lines in the calculation in order to avoid having a singular,noninvertible,matrix Ybus. Electrical distance is not the only distance measure we can use. When defining this measure, we can make other considerations, some of which are described below. Once this distance measure has been defined, we need to find the partition (that is, the assignment of nodes to zones) that minimizes the aggregate intra-zonal distance and, in the cases where the number of zones has not been determined, maximizes the inter-zonal one. Mixed-Integer Programming (MIP) can model this as an optimization problem. However, solution times can be unmanageable even for small-sized networks [9],and,therefore, this technique is difficult to apply to real-size systems. Other approaches include the application of spectral partitioning [9], a technique that is widely used in image segmentation. This method is based on the computation of the eigenvalues of the similarity matrix (which expresses the distance between variables). These eigenvalues are then used to reduce the dimensionality in the attributes before

applying a clustering algorithm. Alternatively, other authors identifY clusters of nodes based on electrical distance using classical clustering algorithms such as K-means or K-medoids [10]. This approach has the advantage of being efficient over large sets of data and constitutes the basis of the proposed technique. However, if network partition is established solely on the basis of electrical distance, all considerations other than network topology are disregarded. No information about the operation of the system, the placement of generation and demand, or the capacities of lines is taken into account. More importantly, there is no guarantee that frequently congested lines will not belong to the same partition, potentially leading the planning process to miss some important reinforcement needs. In addition, using electrical distance can sometimes result in partitions that are in conflict with the intuitions that TSOs can derive from geography alone. The approach we propose and implement in e-Highway2050 is based on combining the information contained in electrical and geographical distances calculated, and incorporating constraints related to the representation of critical branches in the reduced network model. In addition, we propose a heuristic algorithm that builds a partition that will be used as the starting point of the clustering algorithm. This initial partition has the added interest of showing the simplest partition that is able to maintain the explicit representation of all the transmission lines labeled as critical. IV.

NETWORK PARTITION ACCORDING TO CRITICAL BRANCHES

As explained above, our contribution differs from the approach in reference [10] in three main aspects: The partitions are built around the identified critical branches instead of using electrical distance alone. We propose a heuristic algorithm that builds an initial partition based on these critical branches. Instead of using electrical distance, our reduction is based on a composite distance that aggregates additional information that is relevant to the transmission expansion problem (in the case study, this means geographical and electrical distances). The proposed technique is articulated in three steps: Identification of critical branches, Initial network partition, Refinement of the initial network partition using a modified version of K-medoids. A.

Identification of critical branches

As mentioned above, the most relevant transmission lines for the operation of the system will be labeled as critical branches. In addition, some other critical branches will be identified by the analysis of operation scenario data. We propose several criteria for the identification of these branches: Average flow above a threshold percentage of its capacity, which captures the utilization of a transmission line. Even if a line supports a high average flow, it should not be considered a network element to represent explicitly unless the line is already congested. In addition, the

economic impact of congestion affecting different lines can be very different. The following indicators have been chosen to complement average flow as a variable to identifY critical branches and reveal the main congestion paths in the network. We propose to use the proportion of operation hours where the line is congested. If this proportion exceeds a certain threshold, the line is considered a critical branch. In order to select the branches related to congestion with the highest economic repercussion, we propose to use indicators of congestion severity. Two alternatives are proposed: The value of the dual variable of the capacity limit constraints considered when solving the power flow, weighted across periods and scenarios. The difference between the nodal prices at the extremes of the line weighted with the line flow across periods and scenanos. B.

Distance definition

We propose to base network reduction on a composite distance that is calculated as a weighted average of several individual distances. In this way, we can incorporate additional considerations into the analysis. For instance, including geographical distance ensures that geographically compact partitions are favored. Other economic considerations can be taken into account in the definition as well, such as average Locational Marginal Prices (LMPs), so that nodes with similar prices have a higher probability of being clustered together. Alternatively, we can use Power Transfer Distribution Factors (PTOFs) with respect to 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. Although other definitions, as seen above, are possible, in the case study we favor electrical distance combined with a geographical one. Electrical distance captures the structure of the network, while combining it with the geographical one ensures that geographically compact partitions are obtained. LMPs are to some extent considered in the definition of critical branches, given that frequently congested transmission lines related to relevant congestion from an economic point of view tend to have large differences between the LMPs of their extreme nodes, and, therefore, to be selected as critical branches. It is also important to note that only the highest voltage level in the network is taken into account in the partition mechanism, in this case, 380-400 kV, to avoid distorting the electrical distances. The remaining lower voltage nodes are assigned to the geographically closest partition. C.

Initial network partition

The heuristic algorithm builds an initial parlltlOn by starting with a single zone that is potentially split as many times as critical branches have been identified: If both extremes of a critical branch belong to the same zone,this zone is split in two new zones. The nodes that belong 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 represented explicitly in the partition and incorporates all the

information that has been considered relevant for the composite distance calculation. This initial partition is subsequently refined by a clustering algorithm. D. Refining the partition Most of the few works that tackle network partItIon explicitly apply K-means or K-medoids to cluster the nodes into zones according to electrical distance [10, 11]. We use a modified version of K-medoids that avoids assigning both end nodes of a critical branch to the same zone. K-medoids refines the partition by applying the following steps iteratively [12]: The algorithm chooses a representative node for each zone, the medoid, which presents the lowest average distance to the rest of nodes within the zone. The algorithm 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. This is exactly equivalent to the better known K-means algorithm except in the selection of the representative node. In the case of K-means the representative is calculated as the mean of the nodes in the partition. On the contrary,K-medoids chooses the node with the lowest average distance to the remaining nodes in the partition. The medoids calculated in this way have the disadvantage of often being relatively small,not particularly remarkable nodes. Thus, we propose an alternative criterion for the definition of the final representative node of each zone: the most connected node. We define it as the node that has the highest connection capacity to the rest of the network. This guarantees that the representatives will be relatively important nodes in the network, such as large power plants or highly connected substations. V.

CALCULATION OF NETWORK PARAMETERS

Once the zones in the reduced network have been computed, we need to calculate the parameters that describe the corridors among them. Classical network reduction techniques divide the network into internal buses, external buses and boundary buses connecting external and internal buses. They assume that the internal network is known, the external network is remote and has a limited influence on the operation of the internal network and that the number of boundary buses is small compared to the number of internal buses [13]. We propose to follow the approach in reference [14], where the authors define inter-zonal corridors as the equivalents of the sets of lines connecting pairs of zones in the nodal network model. The reactances of the inter-zonal corridors can be calculated solving a system of linear equations that makes use of the PTOF matrix. Contrary to what happens in other reduction approaches, the proposed technique minimizes errors in inter-zonal flows for multiple scenarios simultaneously. The PTOF matrix of the zonal network can be expressed as a function of the aggregate power injections in the zones and the inter-zonal flows, which are both known from the nodal network. By operating these expressions, the PTOF 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 assignment 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. Reference [13] finds these optimal reactances that result in the smallest Euclidean distance between the inter-zonal flows in the original nodal network and those in the reduced one. We develop a method to calculate the capacIties of the inter-zonal corridors. The idea is that capacities of inter-zonal corridors must allow the maximum transfer of power among zones than may occur in the nodal network. In order to calculate this maximum transfer of power among zones, several situations must be considered. We select several base operation situations snapshots. These snapshots represent stressed operation situations for the system. Then, to compute the maximum transfer of power, we simulate the increase of transfer power among zones over the selected snapshots. We use the nodal PTDF matrix and increase generation and demand of these zones until a nodal network limit is reached. This maximum transfer of power is, then, simulated in the zonal network using the zonal PTDF matrix. Finally, the capacities of the inter-zonal corridors are determined so that they allow all the power flows provoked by these transfers of power. VI.

CASE

STUDY

addition to them, 17 other special elements, namely PSTs, have been identified and incorporated to the list of critical branches.

2

C.

The indices described in IV.A were calculated across 10 different scenarios with one time horizon, where the hourly operation is simulated for a year. We have calculated the indicators for critical branches described in IV. In this case, 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 are not identical, they point to the same relevant corridors. The results can be seen in Figure 3. The case study presented in this paper uses the top 100 most relevant critical branches selected by the average flow criterion. However, as explained, the selection of critical branches is not very different when using other criteria. In

10

,,'

12

Nodal price difference

Figure 4 displays the initial partition,which is defined with only 24 zones. This partition represents a good starting solution for the K-medoids algorithm. x1d'

1.

Identification of critical branches

!

Initial partition taking into account critical branches

A.

B.

6

Figure 3. Critical branches (blue lines) according to the ditlerent proposed criteria: average f10w, congestion, marginal severity and nodal price difference.

2

The aggregate distance between nodes is calculated based on electrical and geographical distances. The electrical distance considered follows the definition in section III. There were some outliers in the distribution of distances, which were identified with nodes that are poorly connected to the rest of the system. These cases can lead the algorithm to create partitions composed of a single node. As this is not a desirable situation, all the outliers that were not connected to a critical branch were not considered in the clustering algorithm. They are subsequently assigned to the geographically closest zone.

4

Congestion

Marginal severity

The proposed network partition technique has been applied to a case study based on the French and Spanish very high­ voltage transmission networks and composed of 542 nodes (380-400kV). The target number of partitions is established as 50 in order to keep a one-order-of-magnitude reduction in terms of number of nodes. Distance calculation

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