Market Coupling as the Universal Algorithm
to Assess Zonal Divisions
Grzegorz Oryńczak, Marcin Jakubek, Karol Wawrzyniak, Michał Kłos
National Centre for Nuclear Research, Świerk Computing Centre
Otwock-Świerk, Poland
Karol.Wawrzyniak@fuw.edu.pl
Abstract— Adopting a zonal structure of electricity market
requires specification of zones’ borders. In this paper we use
social welfare as the measure to assess quality of various zonal
divisions. The social welfare is calculated by Market Coupling
algorithm. The analyzed divisions are found by the usage of
extended Locational Marginal Prices (LMP) methodology
presented in paper [1], which takes into account variable
weather conditions. The offered method of assessment of a
proposed division of market into zones is however not limited to
LMP approach but can evaluate the social welfare of divisions
obtained by any methodology.
Index Terms— Flow Based Market Coupling, Power System
Economics, Social Welfare

I.

INTRODUCTION

The energy market in Europe is undergoing a process of
transformation aimed at integration of national markets and
making better use of renewable generation sources. The
market structure used in many countries, mostly due to
historical reasons, is the uniform pricing, in which there is a
single price of energy set on a national market for each hour
of a day. In spite of its apparent simplicity, such an approach
has serious disadvantages. The equilibrium set on the market
does not take into account safety requirements of the grid.
Hence, (i) the single-price equilibrium set on the market
(energy exchange) is frequently unfeasible, (ii) the system
operator has to perform costly readjustments, (iii) costs of
supplying the energy differ between locations, but they are
not covered where they arise.1 With introduction of other
forms of market, congestion costs are mitigated and the price
on the market reflects the true costs of supplying energy to
different locations in a more adequate way.
Hitherto, the explicit type of the future pan-European
energy market remains an open question as the Third Energy
Package does not specify the design precisely, but the two
most popular approaches towards which national markets
evolve are nodal and zonal pricing. The nodal pricing model
is currently used in, among others, the US and Russia. Zonal
pricing has been introduced in the Nordic countries as well as
in Great Britain. This type of pricing is gaining in popularity
across the Europe, and will be the main subject of this paper.
Zonal market, which can be thought of as a compromise
between simplicity of uniform structure and accuracy of
1

For example, in Poland in 2011 the cost of the balancing market
readjustments amounted to more than 3% (>250 million EUR) of the overall
costs of production (source: URE/ARE S.A.).
____________________________________________________________
This work was supported by the EU and MSHE grant nr POIG.02.03.00-00013/09. The use of the CIS computer cluster at NCBJ is gratefully
acknowledged.

nodal one, introduces differentiation of prices between
regions with distinct costs of supplying energy, but it
maintains the transparency which the nodal market lacks. The
grid is divided into geographical regions (zones), each having
a separate market for the energy with possibly different price.
Market Coupling (MC) algorithm is used to control interzonal power flows and to calculate prices in zones given
those flows. This way, under presumption that the zones were
chosen so that frequently congested lines are on their borders,
the equilibrium on zonal markets will take into account
transfer limits on those critical lines. The need for additional
congestion management is thus minimized, with most of the
task being performed by the MC mechanism. Of course,
small adjustments of equilibrium to satisfy limits on intrazonal lines might be necessary, but they are expected to be
less costly than the adjustments on a corresponding uniform
market.
Still, there is no consensus in the literature with respect to
methodology of identification of optimal zones’ number and
their borders. The existing methods are mostly based on twostage approach – assignment of some specific values to each
of the nodes and division of the system into regions by
clustering the nodes over those parameters. Among existing
methods, we can distinguish two popular approaches for
choosing the values characterizing nodes.
The first approach is based on nodal prices, called also
Locational Marginal Prices (LMPs) [2,3]. Nodal price
represents the local value of energy, i.e. cost of supplying
extra 1 MW of energy to a particular node - physical point in
the transmission system, where the energy can be injected by
generators or withdrawn by loads. This price consists of the
cost of producing energy used at a given node and the cost of
delivering it there taking into account congestion. Therefore
LMPs separate locations into higher and lower price areas if
congestion occurs between them. The second approach is
based on Power Transfer Distribution Factors (PTDFs). The
procedure starts from identification of congested lines, for
which PTDFs are then calculated. The distribution factors
reflect the influence of unit nodal injections on power flow
along transmission lines, thus grouping the nodes
characterized by similar factors into one zone defines a region
of desirably similar sensitivity to congestions [4].
The main issue that has to be addressed concerning the
ultimate choice of zonal configuration is the inconsistency in
determinants used to separate the nodes and in the criteria
used to evaluate constructed partitions. The two
aforementioned methods derive divisions from a reasonable
assumption that congested lines should become inter-zonal
links, however, the actual shape of borderlines is based on

several different premises used to (i) label each of nodes with
a value (e.g. nodal price) or a set of values (e.g. PTDFs) and
(ii) to group the nodes into geographic areas using a
clustering technique.
Since the methods of division assign the values basing on
data derived from nodal market structure (e.g. LMPs), there is
a justified need for the assessment which evaluates a newly
defined zonal architecture. We can broadly divide evaluation
criteria into those based on safety of the network, which
include, for example, predictability of the flows on intrazonal lines, and those based on economic (market) measures,
like market liquidity, concentration or social welfare
(suggested by [7] among the others). In this article we
calculate and compare social welfare (SW), defined as the
sum of producers’ and consumers’ surplus in the equilibrium
of supply and demand on the zonal market. Specifically, we
use mechanism of Flow Based Market Coupling (FB MC) to
determine the equilibria on each of the zonal markets, taking
into account the limits of power flows between them.2
As a test case of divisions, we use the results of LMP
methodology presented in [1], which takes into account
variable weather conditions and is applied to a simplified
model of Polish grid, and derive the appropriate welfare
measures. The exact methodology of Market Coupling and
calculation of social welfare is presented in the next section.
In Sec. III we describe the model of network used as the test
case. The results derived on it are presented and discussed in
Sec. IV. In Sec. V we conclude and present directions for
future work.
II.

THE METHODOLOGY

The zonal energy market can be represented as a set of
energy exchanges, each governing the trade between
generators and consumers of energy located in a particular
geographic area. Energy transfers between zones are allowed
and are governed by MC mechanism, which takes into
account the constraints of the inter-zonal transmission lines.
In order to determine safe supply-demand equilibria on each
of energy exchanges, the MC mechanism must determine
how the realization of buy/sell bids translate into (i) power
injections/withdrawals in the nodes of the grid and into (ii)
flows on inter-zonal lines. In doing so, the aim of the MC
algorithm is to maximize the social welfare (consumer
surplus + producer surplus + congestion rent) while keeping
the flows in safety limits.
The mechanism of keeping the inter-zonal flows in safety
limits can be of varying level of complexity. We use in our
approach Flow Based MC, which takes advantage of the
Power Transfer Distribution Factors (PTDF) matrix to
determine how a particular injection/withdrawal influences
power flows on inter-zonal lines. This approach is analogous
to a Direct Current Power Flow (DC PF) model of the flows
We use the term “Market Coupling” specifically as the mechanism
governing the exchange between market zones which are managed by one
system operator (for example, the case of a national market divided into
zones, or a common zonal market spanning across more than one country).
Governance of the flows on the borders of two (or more) not integrated
markets is not studied in this paper.

in the grid. As such, FB MC is a significant step-up in
robustness compared to simpler MC mechanisms, for
example the Available Transfer Capacity (ATC) MC, which
limits only the aggregated flow between the zones, without
calculating explicitly the flows on each of the inter-zonal
lines.
We based our implementation of FB MC on the
description of the COSMOS/Euphemia algorithm [5,8],
which was derived for the CWE (Central Western Europe)
energy market.
A. Input of Offers
The main input data for the MC algorithm are the buy/sell
offers of the energy consumers/generators. An n-th offer,
n = 1,…, N, is characterized by affiliation to a zone j,
j = 1,…, J, which we denote by n  Zj and which is derived
from the physical location of the node where the energy will
be injected/withdrawn, and by the volume qn to be traded,
which is coded as a positive amount for a sell bid, and
negative for a buy bid. The algorithm allows partial
realization of an offer and also the use of “triangle” offers, in
which the offer price either decreases (for a buy offer) or
increases (for a sell offer) linearly on the interval [0, |qn|],
thus the offer is characterized by two prices, Pn0 and Pn1, that
is, Pn0  Pn1 for a buy offer, Pn0  Pn1 for a sell offer. In the
equilibrium found by the algorithm, each of the offers can be
either accepted, accepted partially or not accepted at all,
which will be coded by a coefficient An  [0,1]. Thus, an offer
n accepted in percentage An is connected with an
injection/withdrawal of power in the amount of Anqn, with the
highest price in case of “triangle” sell bid (lowest in case of
0
1
0
buy bid) denoted by Pn  Pn  ( Pn  Pn ) An .
B. Flow Calculation
To calculate how the realization of offers affect interzonal flows for a given vector of zonal injections
Q = (Q1,…,QJ)T (the aggregated injections/withdrawals
representing accepted offers in all nodes in a given zone),
which coordinates are given by

Qj 

Aq
n

n

,

(1)

nZ j

we use the nodal PTDF matrix (nPTDF), derived from the
model of the grid, and the Generation Shift Key (GSK) matrix
[6] to obtain the flows along K inter-zonal lines (vector
T
Q  (Q1 ,..., QK ) ) as Q  nPTDF GSK Q .
To construct the GSK matrix we firstly run the MC
algorithm with the flows calculated without the use of GSK as
Q  R nPTDF Q , where R matrix selects the flows along K
inter-zonal lines. The load/generation equilibrium, found for
such constraints, is used to calculate the GSK matrix, which is
then treated as input to the “proper” MC FB algorithm’s run.

2

C. Objective Function and Constraints
Maximization of the social welfare is equivalent to the
optimization problem

 Pn0

F. Redispatch Costs
Since the MC mechanism controls the congestion of only
2
n 1
inter-zonal lines, there is a possibility that system operator
The safety limits on the K inter-zonal lines are described will have to correct the generation profile set on the market in
by their capacities Ck, k  1,..., K , with vector C defined as order to avoid congestion on intra-zonal lines. This process of
C = (C1,…,CK)T. For the vector of zonal injections Q given by redispatch generates additional costs since some of the
(1), the safety constraints on flows in inter-zonal lines, are generation has to be shifted from the cheaper to more
expensive producers. Thus, in order to better reflect the true
characterized by the condition Q  C , which translates to
costs of supplying energy in our social welfare measure, we
correct the amount (5) by estimator of the redispatch costs.
(3)
nPTDF GSK Q  C .
To this end, on the load/generation profile acquired as a
Lastly, we add the balance condition, which states that the solution from MC, we run Power Flow algorithm to calculate
sum of energy bought on a market of zone Zj and imported to flows on intra-zonal lines and we compare them with the
the zone Zj must be equal to the energy sold on this market lines’ capacities. If a line l in zone Zj is congested by amount
and exported from zone Zj, that is,
of ol MW, we add to the cost of redispatch the amount
N

max

A q
n

n



 qn An





nZ j

Pn

Ql 

l  from ( Z j )

.

(2)

Q,
l

(4)

l to ( Z j )

where by from(Zj) and to(Zj) we denote the sets of the interzonal lines along which the energy flows are, respectively,
withdrawing power from and injecting power into zone Zj.
The optimization problem described by objective function (2)
and constraints (3) and (4) is then input to the IBM CPLEX
mixed integer quadratic solver, and the vector of offer
acceptance levels A = (A1,…,AN) is derived.
D. Market Clearing Prices
When the acceptance levels A are found, we can identify
the Market Clearing Prices (MCPs) for each of the zone Zj,
namely MCPj, j = 1,…, J. In general, a clearing price on a
market is defined as any price for which the aggregated
supply and demand (taking into account the import/export
flows) is in equilibrium. Since such price might be not
determined exactly (there might exist a range of prices
satisfying the equilibrium condition), we define MCPj in the
following way, which was chosen to accommodate the nonelastic demand assumption (cf. section III): (i) if demand in
zone Zj was satisfied (for all buy offers in zone Zj we have
An = 1), then we take as MCPj the highest price of a sell offer
accepted at or imported to Zj; (ii) if demand in zone Zj was
not satisfied completely (there exist a buy offer in zone Zj
such that Aj < 1), then we take as MCPj the common price of
buy offers, P , as defined in section III.

ol Pj

max

, where Pj

max

is the highest cost of generation in zone

Zj, to obtain an upper bound on the redispatch costs.
III.

THE TEST CASE

To test our approach on an exemplary case which can be
treated as a relatively close representation of a real energy
network, we used the data on the Polish grid based on the
case file included in MATPOWER distribution [6]. This case
represents the Polish 400, 220 and 110 kV network during
winter 1999-2000 peak conditions.
The system consists of 327 power generators, 2383 buses,
and 2896 interconnecting branches. The quality of the data is
unknown but the fact that the assumed costs of generation are
linear gives a premise that these data are of rather poor
quality. Hence, the analysis should be treated as exemplary,
with no constructive conclusions for the Polish system. We
used this specific case since this is the only available one in
which congestion exists under base case load. Additionally,
we decreased capacity limits of two specific branches in order
to obtain more pronounced influence of congestion on the
MC solutions. A more detailed description of the data can be
found in [1].

We divided the grid taking into account variable wind
generation by using the method presented in [1]. We used
three variations of our methodology, which reflect division
for (i) no wind output, (ii) maximal wind output registered in
the period between years 2007–12, and (iii) so called
E. Social Welfare
consensus clustering, which reflects aggregation of 722
The social welfare in the market equilibrium found by the different divisions made from various weather conditions into
above procedure is equal to
one. In each of the three variants we obtained divisions into 2,
3, 4 and 5 clusters, resulting in twelve divisions to be
0
J

Pn  Pn  

evaluated by the MC algorithm. Each of these 12 divisions
SW 
 An qn  MCPj  2  
was then tested in conditions reflecting average wind


j 1  nZ
(5) conditions in the period between years 2007–12.

 
j

J

J

 ( MCPj  MCPk ) f jk ,
j 1 k 1

where second double sum is the overall congestion rent of the
system operator (fjk indicates power transfer between adjacent
zones j and k).

Specifically, for each division we constructed offers of
wind generators taking the estimated power output for
average wind levels and offering it for a price equal to zero in
order to secure the “sell” of the wind energy, which has
priority over the conventional generation in the system. For
conventional generators the energy available for sell, qn =

3

Namely, the values in Table 1 show the differences between appropriate
levels for the solutions for market divided into 2, 3, 4, 5 zones and the levels
for single-zone solution.

Variant 3
max wind

0

0

-7 144

-3 558

-3 558
3 558
-26 357
-7 147
-7 139
-107 163
5 734
5 052
80 806
-30 943
-30 130
-27 234
-90 622
-103 082
-107 132
59 679
72 952
79 898
-31 460
-30 129
-27 650
-90 645
-103 083
-108 346
59 185
72 954
80 696
5 559
-1 413

3 558
-2 087

total power
demand
[GW]

# generators
357
366
366

24.1
24.6
24.6

147
147
147

97
22
21

1.6
0.3
0.3

9
0
0

0.5
0
0

2000
147
147

yellow

2325
2234
1592

23.0
21.8
15.3

357
343
218

24.1
23.3
18.9

147
147
129

22
22
21

0.3
0.3
0.3

0
0
0

0
0
0

2000
2000
178

75
166
809

1.3
2.5
8.9

9
23
148

0.5
1.3
5.6

2000
2000
159

1477
1286
1592

14.8
12.6
15.3

255
163
218

12.1
15.5
18.9

152
129
129

848
948
734

8.1
9.2
7.6

102
180
139

11.9
7.8
5.1

129
159
159

75
166
75

1.3
2.5
1.3

9
23
9

0.5
1.3
0.5

2000
2000
2000

22
22
21

0.3
0.3
0.3

0
0
0

0
0
0

2000
2000
2000

total output
[GW]

# nodes

23.0
24.3
24.3

2 zones

2325
2400
2401

green

3 zones

Market
Clearing
Price
[PLN/MWh]

Table 1. Social welfare, redispatch costs and corrected SW in relation to
single-zone market from for grid divisions into two, three, four and five
zones done in 3 variants - no wind/consensus/maximal wind.

purple

The results summarized in Tab. 1 show that, as was
expected, the (uncorrected) SW in the case of a single-zone
market turned out to be the highest, since no congestion
constraints are then put on the market solution. However,
high redispatch costs associated with correction of this
solution lead to the lowest corrected SW for single-zone
market. The best division (with the highest SW corrected by
redispatch) are related to ‘max wind’ variant for 3 clusters. In
turn, the worst results are obtained for 2 clusters divisions in
the ‘consensus’ and ‘max wind’ variants. One can notice that
the corrected SW rises while redispatch costs drop with
increasing number of clusters up to 4 (for ‘no wind’ and

-1 585

Variant 2
consensus

yellow

Since, as yet, there is no widely accepted methodology for
choosing the exact number of zones into which market should
be divided (although there have been some attempts, cf. [11]),
in our study we assumed the range of the three
aforementioned variants of divisions (no wind, maximal
wind, consensus) to, respectively, 2, 3, 4 and 5 clusters,
which yielded 12 division cases. Then we analyzed each of
them separately and compared the results of social welfare in
those 12 cases. In Tables 1&2 we characterize the results
quantitatively, while Figure 1 delineates the geographical
placement for divisions into 3 and 4 zones. Tab. 1 shows
differences between SW, redispatch costs and SW corrected
by redispatch levels calculated for zonal divisions with
respect to single-zone market as a reference point.3 In the
case of single-zone market, SW is calculated from the MC
solution in which there is only one zone, thus no congestion
management is embedded in market mechanism and all is
done by redispatch. SW in such case amounts to 47 470 970
PLN, redispatch costs: 116 589 PLN, SW corrected by
redispatch: 47 354 381 PLN, marginal clearing price: 147
PLN.

Variant 1
no wind

grey

RESULTS & DISCUSSION

4 zones

IV.

Number of
zones / type of
division
SW
2
redispatch
SW corr
SW
3
redispatch
SW corr
SW
4
redispatch
SW corr
SW
5
redispatch
SW corr

purple

From the consumers’ side, since only constant loads at
each bus are available in the data, we assume that demand is
perfectly inelastic, namely, that the loads at each bus are
expected to be covered at any cost. To input such buy offers
to the algorithm, for each nodal load we use an offer with qn
equal to the negative of load, and with the price set at a level
Pn0 = Pn1 = P common for all demand offers. In calculations
presented below we have arbitrarily chosen P  2000 PLN ,
which is greater than any production cost in the data. Since
we are interested not in the absolute level of social welfare
for each proposed zonal division, but in the comparison of the
social welfares obtained across the 12 different divisions, the
choice of the demand bid price has no influence on the
results, as long as this price is equal in every division case
and is higher than all the sell offers’ prices.

‘consensus’ variants) or 3 (for ‘max wind’) and then both
become relatively stable. This can be interpreted as a result of
the most congested lines being taken into account as interzonal constraints into MC mechanism (instead of the costly
redispatch) when the market is divided into 4 (or 3 in case of
‘max wind’) zones. Further increasing the number of zones
does not lead to significant improvement.

green

pmax, where pmax is the maximal output of the power plant, is
offered at a constant price (Pn0 = Pn1), since the costs of
generation in the used case file are linear. That is, we assume
that the generators bid the available amount of energy by the
marginal cost of production.

Table 2. Qualitative data for division of the Polish power grid into 2,3,4
zones. Each cell includes values for no-wind / consensus / max-wind variant
in that vertical order. The colors reflect the areas depicted on Fig. 1.

Looking at Tab. 2, one can notice that usually in a bigger
zone the price is lower, in contrary to a smaller zone, where
the final price is higher than the one for the single-zone
market (147 PLN). Hence, when the bigger zone gets a lower
price, then the total SW increases. (However, one must keep
in mind that we do not include the redispatch costs in the
prices calculated as in Section II.D, thus the link between the
MCPs and SW corrected by the redispatch costs, which can
be treated as the best estimator of market efficiency, is not
straightforward.) Another reason for the SW divergence
between the divisions is that demand in tiny zones, especially
those not having their own generation, is sometimes not fully
satisfied (thus MCPs there are often equal to
P  2000 PLN ). In sum, and in relation to SW levels in Tab.
1, one can see that the division with the highest corrected SW
(‘max wind,’ 3 zones) is the one which has the smallest MCP
in the biggest zone, while the demand is fully satisfied in all
of the zones.
V.

FUTURE WORK

Among the issues that ought to be tackled in the future
research we can distinguish two main subjects.
First, we acknowledge the need for improvement of the
LMP-based clustering algorithm and development of other
partitioning techniques that result in all of the zones being
equipped with their own generation. The partitioning methods
which produce bigger bidding areas are expected to eliminate
the problem of exclusively external supply of energy. Also,
the case of zones which overlap due to interfusion of the
corridors formed along different types of transmission lines
(e.g. 220 kV and 400 kV, cf. top right configuration on
Fig. 1) remains unsolved.
Second, in the zonal approach all nodes in a specific zone
are aggregated into one node. The influence of the power
injected into this zone via transfer through the branches is
estimated by zonal PTDFs (zlPTDF) matrices [6]. The
calculation of zlPTDFs requires some assumption about ratios
between generations/loads and net export which are
expressed by GSK matrix. Thus, MC has to work with
inflexible constraints given by GSK, where certain
proportions between loads and generation are constant.
In other words, GSK has to be given as an input to MC.
Thus, its value has to be guessed a priori before MC
optimization starts. Then, in the optimization process, MC
can select different combination of generations/loads than the
one assumed in GSK, which subsequently forces incorrect
evaluation of constraints.
Thus, as the consequence of the rough estimation of GSK
matrices, the MC algorithm operates on unsound
prerogatives. Both under- and overestimation of power flows
along the transmission lines lead to suboptimal use of the
infrastructure [6]. Thus, the process of deriving reliable
GSK/zonal PTDFs is the central task for enhancement of
zonal market stability and efficiency.

Figure 1. Polish grid division for three (left) and four (right) zones. Results
for no-wind, consensus and maximal wind division variants are shown in the
top, middle and bottom row, respectively. Arrows indicate direction and
magnitude of energy transfers between zones in gigawatts.

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