DEFINITION OF SINGLE PRICE AREAS WITHIN A REGIONAL
ELECTRICITY SYSTEM
Luis Olmos, Ignacio J. Pérez-Arriaga
Instituto de Investigación Tecnológica, Universidad Pontificia Comillas
Madrid, Spain
[email protected]
Abstract – The scarce transmission capacity both in local and regional markets should be allocated to the agents
that bid for it based on the application of a coordinated
market-based method. Defining Single Prices Areas
(SPAs) is central to the implementation of any zonal congestion management scheme. If accurate enough SPAs
cannot be defined, we would have to resort to computing a
single energy price for the whole system or, alternatively,
implementing a system of nodal energy prices.
This article presents a novel method for the computation of single price areas that is based on the impact that
transactions between grid nodes have on the flows of the
lines in the system that are likely to get congested. Transactions within a SPA should have a negligible impact on
the flows of these lines whereas transactions taking place
between two or more different areas should significantly
affect these flows. Nodes have been classified into areas
using different clustering techniques.
Following the description of the method, we provide the
results corresponding to its application to the definition of
SPAs both in the Iberian Peninsula and in the system
comprised of France, Spain and Portugal. Finally, we
outline the characteristics of an improved version of the
method presented here which is aimed at reducing the
error committed when computing a zonal instead of a
nodal dispatch.
Keywords: Congestion management, energy prices,
clustering techniques
1 INTRODUCTION
A Single Price Area (SPA) is a set of nodes that is
considered as a single one for congestion management
purposes. Differences in energy prices among the nodes
within a SPA are only related to losses. Defining Single
Price Areas is central to the implementation of any
zonal congestion management scheme. Nodal pricing is
the only reasonable scheme in those systems where
congestion plays an important role and accurate enough
SPAs do not exist.
Local markets can manage the dispatch of energy
within a SPA independently from the rest of the system.
Thus, the identity of agents involved in commercial
transactions taking place within each area, as well as the
amount of power transacted and the price paid for this
power may be kept confidential. Lastly, managing congestion taking place on the border between SPAs is
certainly far less complex than considering the whole
system grid.
A distinction must be made between sporadic and
systematic congestions. Sporadic congestions occur
rarely and are not relevant for the outcome of the market. Systematic congestions occur more frequently and
affect significantly the economic dispatch in a country
or region. Strictly speaking, all congestions in the system should be managed using efficient coordinated
market mechanisms. In practice, leaving authorities
within each area in charge of solving sporadic congestions may be acceptable.
The remainder of this paper is structured as follows.
Section 2 analyzes from a conceptual point of view the
problem of defining SPAs. It includes a description of a
novel method for the identification of SPAs. Afterwards, section 3 presents some numerical results corresponding to a preliminary division into SPAs both of
the Iberian Peninsula and of the system comprised of
France, Spain and Portugal. Section 4 outlines the main
characteristics of an improved version of the method
proposed here, which aims to minimize the decrease in
the social benefit resulting from the application of a
zonal instead of a nodal dispatch. Finally, section 5
draws some conclusions and discusses some lines of
future research.
2
CONCEPTUAL ANALYSIS OF THE
PROBLEM
A transaction taking place between nodes belonging
to the same SPA should not affect the flow on the congested lines or corridors. Different patterns of use of the
network may result in different congestion patterns.
Thus, boundaries of SPAs may vary from one scenario
to another. Any correctly defined SPAs should be valid
for any possible network configuration and under any
possible system conditions.
If a linear model is considered acceptable to represent the way power flows over the network, one may
conclude that all the power transactions between two
SPAs must have the same impact on the flows of congested lines. Therefore, all the nodes in a SPA may be
represented by just one. Two transactions of the same
size starting and ending in the same two areas are then
totally equivalent.
Two nodes belong to the same SPA if, and only if,
two transactions taking place between each one of these
two nodes and any other one produce the same flows on
1
16th PSCC, Glasgow, Scotland, July 14-18, 2008
Page 1
the congested lines. This must be true under any set of
operating conditions.
Thus, provided SPAs actually exist, nodes belonging
to each SPA may be identified as those such that all
elementary transactions taking place between them and
a common reference node produce the same flows on
the congested lines. This may be assessed using the
Power Transfer Distribution Factors of the flow through
the congested lines with respect to these elementary
transactions. This is the approach that we have followed
to define SPAs. Other authors in the academic community have proposed similar algorithms to identify these
areas. In [1] a system is divided into several areas according to how similar the PTDFs obtained for the different nodes are. Afterwards, they re-dispatch generation within each area to manage congestion in the grid.
Generators picked up to participate in re-dispatch are
those whose PTDFs are large and different from those
obtained for any other node, i.e. those that are close to
the congested lines. Authors in [2] identify price areas
using the PTDFs of the flow through the interconnection lines between control areas with respect to transactions.
Unfortunately, defining clear-cut SPAs in meshed
systems may not be possible. Using a simplified representation of the system grid may require accepting some
margin of error in our calculations. Therefore, defining
SPAs may probably result in some efficiency losses in
the management of congestion.
The next paragraphs present the method that has
been applied in our study. This method classifies the
nodes in a system according to the flows that elementary transactions taking place between them and a reference node cause on the congested lines. As discussed
above, this method is valid only when the fraction of the
scheduled intra-area transactions that flows through
congested lines is not significant.
2.1 Description of the proposed method for the computation of SPAs
We have used clustering techniques to classify nodes
into zones. The methodology proposed here makes use
of the topology of the grid. We must minimize the use
that intra-area transactions make of the lines that are
likely to be congested. In addition, nodes in an area
must be electrically connected among themselves.
We have computed Power Transfer Distribution
Factors of the flows over the potentially-congested lines
with respect to elementary transactions between each
node in the system and the reference node. Once this set
of PTDFs has been computed, estimating the impact of
a transaction between any two nodes in the system on
any potentially congested line is immediate. We only
need to compute the difference between the PTDFs
corresponding to the injection and withdrawal nodes in
the transaction. Nodes belonging to the same area have
the same elementary PTDFs since a transaction between
them does not affect the flows on the congested lines.
Clustering algorithms produce groups of samples
minimizing the distance (difference) among members of
16th PSCC, Glasgow, Scotland, July 14-18, 2008
the same group while maximizing the distance between
elements belonging to different groups. Two different
algorithms have been used: KSOM and kmeans, see [3,
4]. The number of groups, or clusters, is an input to the
model. We do not know in advance which one is the
ideal number of groups (areas). Therefore, several classifications corresponding to different numbers of clusters must be obtained.
Due to the fact that most clustering algorithms are iterative processes, an initial solution (classification)
must be provided. The initial solution is used by the
algorithm as a starting point in the iterative process. The
algorithm rarely arrives at the true optimum and the
final solution obtained depends on the starting point
considered. For this reason, several experiments have
been conducted for any given number of clusters or
areas. Thus, we can assess the stability of results with
respect to the choice of the starting point.
All nodes within any of the areas must be electrically
connected among themselves. Prices in nodes that are
not electrically close within a congested system cannot
be thought to evolve jointly with each other. Thus,
nodes in an area that are not electrically connected to
the remaining nodes have been assigned to another area.
Afterwards, we computed the overall error for each
division into areas. Intra-area transactions should not
modify the flow on congested lines. Therefore, the error
corresponding to an intra-area transaction
may be
defined as the amount of power produced by transaction
that flows through congested lines. The overall error
committed for a given classification has been characterized using the maximum and average transaction errors
over all the possible transactions.
After computing the error corresponding to a classification, we assessed the performance of this classification according to several criteria. The list of considered
criteria follows:
• The maximum transaction error committed for the
whole classification must be as small as possible.
• The average transaction error committed for the
whole classification must also be as small as possible.
• The number of nodes within each area must be high
enough.
• The division into areas obtained must be compatible with the existing political and organizational
structures.
Finally, the classification that ranks highest, taking
into account at the same time the above mentioned criteria, must be implemented.
2.2 Clustering techniques applied
Two different clustering algorithms have been used
in the study: the Autoregressive Kohonen Maps
(KSOM) and the kmeans algorithm. These algorithms
are of two different types. When dividing a set of samples into several groups or clusters using the KSOM
method, we may define a higher number of cells (elementary groups of similar samples from which clusters
are then created) than the number of clusters we want to
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create. Thus, we say that the KSOM algorithm is a
flexible method. On the other hand, the number of cells
employed in the kmeans algorithm must be the same as
the number of clusters we want to define. Thus, we say
the kmeans algorithm is a rigid one.
The KSOM algorithm represents on a twodimensional cell map the samples to be classified into
areas. The number of cells in the map is an input to the
model. A color code is used to specify the distance
(difference) that exists between any two samples. When
several classification variables exist (as it is the case
here, where each variable corresponds to the flow on a
different congested line) separate maps can be drawn
for each of them. In the latter maps, colors represent the
value that the corresponding variable adopts in the different samples.
From these maps it is possible to identify groups of
samples that are similar among themselves but are quite
different from the rest of the samples. The number of
cells in the map may coincide with the number of
groups of similar samples identified or the number of
clusters we want to obtain. In this case, each cell corresponds to a different cluster (area). Otherwise, we have
to group the cells into clusters.
3 IDENTIFICATION OF SINGLE PRICE
AREAS IN THE IBERIAN PENINSULA AND THE
SYSTEM SPAIN-FRANCE-PORTUGAL
We have applied the approach that was proposed in
the previous section to define SPAs within two real
systems: the Iberian Peninsula and the sub-region including France, Spain and Portugal. The classification
variables used in our analysis are the flows produced by
elementary transactions on a set of lines identified by
the Spanish TSO as belonging to a corridor that is
likely-to-be-congested. We were not able to use other,
more sophisticated, classification variables, like nodal
energy prices, since the required economic information
for the two systems was not available to us.
3.1 Results for the identification of areas in the Iberian
Peninsula
According to some first estimates by the Spanish
System Operator, three lines may be systematically
(frequently) congested in the Iberian system presently.
All of them are cross-border lines, i.e. they cross the
border between the Spanish and the Portuguese systems. A list of these lines follows:
• Line between the 220 kV nodes named ‘Aldeadávila’ (Spain) and ‘Bemposta’ (Portugal).
• Line between the 220 kV nodes named ‘Aldeadávila’ (Spain) and ‘Pocinho’ (Portugal).
• Line between the 220 kV nodes named ‘Saucelle’
(Spain) and ‘Pocinho’ (Portugal).
All the nodes are part of one of the main electrical
corridors between the Spanish and the Portuguese systems. The three lines have a nominal capacity of
380MW.
16th PSCC, Glasgow, Scotland, July 14-18, 2008
We have applied the proposed methodology to a
snapshot corresponding to the winter peak load of the
year 2004. The snapshot is comprised of 1200 nodes.
Each node has been assigned a label representative of
the Spanish province or the region in Portugal where it
is located. The best possible classifications of the system nodes into 2, 3 and 4 SPAs have been computed
applying the method presented in the previous section.
Next, we briefly describe the main characteristics of
these classifications.
3.1.1
Classification into 2 SPAs
We have classified the nodes of the Iberian peninsula
into two groups of them or SPAs. Several classifications
have been obtained. As mentioned above, two different
clustering algorithms have been applied: KSOM and
kmeans. Maps of 2, 12 and 25 cells have been produced
using the KSOM method. If the number of cells is
higher than the number of specified areas, then cells
must be grouped into areas. Several experiments have
been conducted for each combination of a clustering
algorithm and a number of cells to be explored.
In order to determine which one of the explored classifications was the most efficient one, we compared
them according to the maximum error committed for a
transaction in each of these classifications and the average transaction error1. Based on this comparison, we
concluded that the best performing classification is the
one where SPAs are the same as countries. The maximum transaction error committed in the chosen classification is 54.1% while the average transaction error is
between 2 and 3%.
3.1.2
Classification into 3 SPAs
This subsection presents the results obtained when
dividing the Iberian Peninsula into three areas. All of
the classifications but one include at least one area that
is comprised of less then 5 nodes. According to the
evaluation criteria employed, this is not acceptable.
Hence, the remaining classification was chosen as the
best possible division of the Iberian Peninsula into three
areas. Now we list the nodes included within each SPA:
• Area 1 includes all the Portuguese nodes.
• Area 2 includes 27 nodes in the western part of
Salamanca, 21 nodes in Zamora, 4 in Salamanca
and 3 in Valladolid. All of them in Spain.
• Area 3 includes the remaining Spanish nodes.
The overall average transaction error committed for
this classification is 1.4%. This means that, on average,
1.4% of the power corresponding to an intra-area transaction flows through the congested lines. The maximum
intra-area transaction error amounts to 54.1%.
1
The error committed for a transaction within a certain classification is defined as the difference between
the flows produced by this transaction on the congested
lines and those produced by a typical transaction whose
origin and end are located in the same pair of SPAs
according to this classification.
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3.1.3
Classification into 4 SPAs
According to the aforementioned criteria, the best
classification of the nodes within the Iberian Peninsula
into 4 areas that we were able to obtain is the following
one:
• Area 1 includes 133 Spanish nodes (39 nodes located in La Coruña, 4 located in Leon, 30 nodes in
Lugo, 47 nodes in Orense, 13 in Pontevedra) and 1
node in the Northwest of Portugal. The total number of nodes in Area 1 is 134.
• Area 2 includes 16 nodes located in the centre east
of Portugal, 30 in the centre west, 3 in the north
west, 8 in the north and 3 in the south west. The total number of nodes in Area 2 is 60. All in Portugal.
• Area 3 includes 4 nodes located in Salamanca, 27
in the west of Salamanca, 2 in Valladolid and 21 in
Zamora. The total number of nodes in the area is
54. All in Spain.
• Lastly, area 4 includes the remaining 947 nodes of
the Spanish system.
The average transaction error committed for this
classification is 0.9% while the maximum error is
54.1%.
3.1.4
Final choice of a division of the Iberian Peninsula into SPAs
We must determine which one is the optimum number of SPAs to define. One must compare the best classifications obtained when dividing the system into 2, 3
and 4 areas. These 3 classifications have been assessed
in the light of the criteria outlined in section 2.
The maximum error seems to be unaffected by the
number of areas considered while the larger the number
of areas defined the smaller is the average error. However, the average transaction error is quite small even in
the worst case. On the other hand, as far as the practicability of each division into areas is concerned, defining
two areas that are coincident with countries certainly is
far less problematic than establishing three or four areas. Therefore, in the context of the Iberian electricity
market, we recommend dividing the peninsula into two
areas, each one coincident with a country. The maximum transaction error for the classification into areas
that has been finally chosen seems to be unacceptably
high. Now we briefly describe a possible solution to this
problem that has not been implemented in the research
work leading to this paper.
In order to reduce the maximum transaction error
committed, one may think of dealing independently
with those transactions for which the error committed is
highest. These transactions may be called special. Both
transactions internal to an area and transactions between
different areas may qualify as special. Special transactions must receive specific treatment in the congestion
management mechanism. They must be dealt with as
transactions taking place between two specific nodes
instead of two areas (or instead of transactions internal
to an area that can be ignored).
16th PSCC, Glasgow, Scotland, July 14-18, 2008
Identifying those transactions that deserve specific
treatment involves setting a threshold for the error
committed when considering a transaction as a typical
inter-area or intra-area one. If, once a system is divided
into SPAs, the difference between the set of flows produced by a transaction T on the congested lines and
those flows characteristic of a transaction between the
same two SPAs is above this threshold, transaction T
must receive a specific treatment. Specific PTDFs of the
flows on the congested lines with respect to transaction T must be computed. Obviously, this is at odds
with the application of a zonal congestion management
system. Hence, dealing individually with certain transactions is only possible if there are a small number of
these transactions. We must compute the optimum
trade-off between the error committed when dividing
the system into areas, which is higher the higher the
threshold is, and the number of special transactions,
which is higher the lower the threshold is.
3.2 Results for the identification of areas in the system
France-Spain-Portugal
Now we investigate the division into areas of the system comprised of France, Spain and Portugal. This
time, not only the congested lines between Portugal and
Spain must be taken into account but also those between
the Spanish and the French systems.
The representation of the Iberian system considered
here is simpler than the one studied in the previous
subsection. In fact, the congested corridor between
Spain and Portugal is modeled using two lines instead
of three. The whole system is comprised of 1727 nodes.
The lines deemed to be congested are: the 220KV
line between Aldeadávila (Spain) and Pocinho (Portugal), the 220KV one between Saucelle (Spain) and
Pocinho (Portugal), the 220 KV line between Biescas
(Spain) and Pragnere (France), the 220KV line between
Vic (Spain) and Baixas (France) and the 220KV line
between Hernani (Spain) and Cantegrit (France). We
could have also considered the 400KV lines between
France and Spain. However, results obtained when
considering all cross-border lines should not differ from
the ones presented here, since 400KV and 220KV
cross-borderlines between France and Spain have been
built in parallel.
The number of SPAs in the system is an input to the
method used to identify them. We have explored the
possibility of defining 3, 4 and 5 areas. The remaining
of this subsection is devoted to presenting and discussing the results obtained in each case. Finally, we choose
the best possible division of the system into SPAs.
3.2.1
Classification into 3 SPAs
The best possible classification of this system into 3
areas is depicted in Figure 1. The maximum and average transaction errors for this classification are 48.5%
and 3.7%, respectively.
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3.2.2
Classification into 4 SPAs
Figure 2 is a graphical representation of the best possible classification of this system into 4 SPAs that we
have been able to find. Average and maximum transaction errors are 3.4% and 47%, respectively.
A1
A3
A2
Figure 1: Division into three SPAs adopted for the system
comprising Spain, France and Portugal
A1
A4
A3
A2
Figure 2: : Division into four SPAs adopted for the system
comprising Spain, France and Portugal
3.2.3
Classification into 5 SPAs
Finally, Figure 3 shows the best division of this system into 5 areas according to the method proposed for
their identification. Average and maximum transaction
errors are 3.2% and 47%, respectively.
A3
A2
A4
A1
A5
Figure 3: : Geographical distribution of the 5 SPAs that have
been defined in the system that is comprised of the French,
Spanish and Portuguese ones
3.2.4
Final choice of a division of the system comprised of France, Spain and Portugal into SPAs
At this stage of the process we are in the position to
decide whether to divide the system into 3, 4 or 5 areas.
We must compare the best classifications obtained in
these three cases. According to the results provided in
the previous subsections, the maximum intra-area transaction error committed decreases with the number of
areas. However, this reduction is very modest. The
division into 4 areas probably is the one that ranks
highest. Specifically: 1) the maximum transaction error
for this classification is the lowest one, 2) its average
transaction error is very close to the lowest one and 3) it
is easier to implement than the division into 5 areas.
4 DESCRIPTION OF AN IMPROVED
VERSION OF THE PROPOSED METHOD
Any division of a system into Single Price Areas
must be assessed with respect to two main features:
• Simplicity of implementation.
• Efficiency in the operation of the corresponding
zonal system.
The level of compliance with the first objective can
be measured in terms of the number of defined SPAs
and their identity (whether they can be assimilated to
existing political or organizational structures). The level
of compliance with the second one may be measured in
terms of the welfare losses incurred when implementing
the proposed division into areas with respect to the most
efficient solution that is possible (the nodal pricing
scheme).
Provided the defined SPAs are of enough size, we
could say that, for any number of SPAs, the best division is the one that most closely approximates the outcome of a nodal pricing scheme. Thus, considering a
certain snapshot of a system, we may assess how efficient a division into areas is by following the process
described next:
1. We compute the optimal dispatch for the system.
This needs to be done only once for each snapshot.
2. We then compute the outcome of the zonal dispatch
assuming certain areas and the corresponding set of
PTDFs for the flows on the likely-to-be-congested
lines with respect to the transactions between areas.
3. Finally, we compute the efficiency losses incurred
for this division into areas. Losses may be of two
types:
a. The difference between the total system
welfare in the reference case and that corresponding to the zonal dispatch.
b. The economic cost associated with the infeasibilities resulting from the zonal dispatch. This may be approximately measured as:
ECI = ∑ φexc ,l γ l
(1)
l
where ECI is the economic cost of the infeasibilities, l
is the set of congested lines, φexc ,l is the flow on line l in
16th PSCC, Glasgow, Scotland, July 14-18, 2008
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excess of the capacity of the line and γ l is the dual variable of the binding constraint that limits the flow on line
l.
Total welfare losses may be obtained as the sum of
losses of both types.
We now have a method to assess the efficiency of a
classification. A heuristic process similar to the one
described in [5] could be employed to search for the
most efficient classification of the system nodes into a
certain number of areas. In this process, nodes would be
transferred from one area to another, one at a time, and
efficiency losses would be evaluated for both the original and the final classifications. In this way, we could
assess whether a certain change in the division of the
system into SPAs results in a more efficient dispatch or
not. A certain initial solution (classification of the nodes
into areas) must be computed somehow. This method is
similar to the one described in [6].
Once we have computed the best possible division of
the system into different numbers of areas, we must
determine how many areas to define. In principle, we
should choose the division into SPAs that achieves the
optimum trade-off between the simplicity of implementation and the efficiency of the dispatch.
5 CONCLUSIONS AND WORK AHEAD
Defining Single Price Areas is central to the implementation of any zonal congestion management scheme.
Coordinated congestion management methods must be
employed to manage congestion on the borders between
areas. SPAs have been successfully defined in some real
life systems like the Nordel market. Many parties favor
the implementation of a mechanism to implicitly manage the congestion caused by cross-border flows in
Europe. Application of this mechanism will certainly be
inefficient unless appropriate SPAs are defined (assuming they exist).
We have proposed a simple method to identify SPAs.
In this method, nodes are classified into areas applying
clustering techniques. The classification variable we
have used is the set of Power Transfer Distribution
Factors of the flows over the congested lines with respect to an elementary transaction between each node in
the system and a common reference node. The only
input to this method is the topology of the system grid.
Other methods may be superior to the one proposed
here when we try to identify SPAs within a meshed
network. However, these more refined methods require
having access to economic information about the energy
bids by market agents or about the variable costs of
these agents. This information will probably not be
made available to the institution in charge of defining
SPAs.
Unfortunately, clear cut SPAs only exist in radial
systems. In any case, there is still room for improvement in the application of the method proposed in this
article. We may single out transactions that cannot be
accurately characterized as an inter- or an intra-area
16th PSCC, Glasgow, Scotland, July 14-18, 2008
transaction. These transactions should be dealt with
separately from the rest by applying, for instance, the
method that has been suggested in section 3.1.4. Apart
from this, contingency scenarios (in addition to the base
case scenario) should be taken into account when computing the set of PTDFs used as the classification variable in the clustering algorithm.
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