Real and reactive power allocation
in bilateral transaction markets
E. DE TUGLIE
Dipartimento di Ingegneria dell’Ambiente
e per lo Sviluppo Sostenibile (DIASS)
Politecnico di Bari
Viale del Turismo, 8, 74100 Taranto
ITALY
G. PATRONO F. TORELLI
Dipartimento di Elettrotecnica ed Elettronica
(DEE)
Politecnico di Bari
Via E. Orabona, 4, 70125 Bari
ITALY
Abstract: - In deregulated power systems, transmission networks are available for third-party access to allow power
wheeling. In such a deregulated environment, the ancillary services are no longer treated as integral to the electricity
supply. They are unbundled and priced separately, and system operators have to purchase ancillary services from
ancillary service providers.
Issues pertaining to costing of ancillary services, and hence appropriate pricing mechanisms for all market
participants to recover the costs, become an important issue for proper functioning of the system. This paper presents a
method able to allocate real power losses and reactive power support to each transaction in a multi-transaction
environment. The approach utilizes a static power system representation in terms of nodal admittance matrix in
conjunction with a classical power flow. With this formulation, it is possible to derive system losses as well as reactive
supply expression as a sum of partial terms due to each transaction.
Key-Words: - Loss allocation, Reactive Allocation, Simultaneous bilateral contracts, Multiple wheeling transactions
1
Introduction
The last decade has seen a worldwide trend towards
restructuring and deregulation of the power industry.
The main objectives of these reforms are achieved
through a clear separation between production and sale
of electricity, and network operations. The energy can be
sold by generation companies through long-term
contracts or by bidding for short-term energy supply at
the spot market [1,2]. Transmission is still a natural
monopoly since the economics of scale are very high,
even if transmission open access has proved to be an
important requirement in deregulated systems. To
guarantee a level playing field for generators and
customers to access the transmission network, the
transmission operator is required to be independent from
other market participants.
The Independent System Operator (ISO) has acquired
a central coordination role and carries out the important
responsibility of providing for system reliability and
security. It manages system operations, such as
scheduling and operating the transmission-related
services. The ISO also has to ensure a required degree of
quality and safety, provide corrective measures when
faced with incidents, and several other functions. In
addition the ISO could also manage market
administration, energy auction and unit commitment
functions in the pool market structure.
To this effect, certain services, such as, scheduling
and dispatch, frequency regulation, voltage control,
generation reserves, etc. are required by power system,
apart from the basic energy and power delivery services.
Such services, which are now commonly referred to as
Ancillary Services (ASs), had all along been part of the
normal electricity supply and were not separated in the
traditional vertically integrated power systems.
However, in deregulated power systems, transmission
networks are available for third-party access to allow
power wheeling. In such a deregulated environment, the
ancillary services are no longer treated as integral to the
electricity supply. They are unbundled and priced
separately, and system operators have to purchase
ancillary services from ancillary service providers.
Issues pertaining to costing of ancillary services and
hence appropriate pricing mechanisms for all market
participants to recover the costs become an important
issue for proper functioning of the system.
Among all ancillary services, electricity markets
identify the supply of losses and reactive power as
crucial factors in system support services. As markets
develop in managing real power transactions, qualifying
and compensating active and reactive power support
service is becoming a more interesting research target,
especially when an increasing number of transactions are
utilizing the transmission system and losses and voltages
become bottlenecked while transferring additional
power.
Different methods are presented in technical literature
in order to share transmission losses [3-10] or reactive
power [11-20] among all participants.
Physical laws governing network flows can have
anomalous and unexpected market implications [17]
thus, real and reactive flows need to be considered in an
integrated approach to perform a correct analysis of
energy markets [21-27]. Following the same purpose, in
this paper a method for allocating real and reactive
power services to individual transactions is developed.
The methodology considers all transactions acting
simultaneously as characterizing the operating point in
nodal voltage terms and evaluated by a usual load flow
code. The system, represented by its nodal current
equations, gives rise to an expression of complex powers
evaluated at the slack bus and reactive generators as a
sum of transaction terms plus a unique network term
depending on transmission system parameters and nodal
voltages. The method allocates real power losses and
reactive power support to each transaction in a multitransaction environment.
2
Bilateral Transaction Framework
Modern electricity markets give the possibility to
exchange power in different ways from the past among
market participants. In fact, if in the past all generated
power was centrally dispatched from generators to loads,
nowadays unbundling and open transmission access
gives the possibility to have direct agreements between
generators and loads. Moreover, this brisk and lucrative
market enables others purely financial actors to
participate in the games. Thus, individual generators can
sell power directly to loads or to a pool or to trading
entities, bringing transactions to assume a double aspect:
both physical and financial.
Trading entities, responsible for financial transactions,
play an important role in modifying power flows
through the network, impacting the physical world. For
such transactions, it is necessary to quantify their
responsibilities in terms of costs associated to the
transmission network usage and losses. Fortunately, in
[2], it is demonstrated that all kind of transactions can
have bilateral generator-to-load (or bus-to-bus)
equivalents. Thus, without loss of generality, we refer
only to physical bilateral transactions aimed to transfer
active power from generators to loads.
For our purposes, as will be clear hereinafter, we need
the following assumptions:
- the market model is characterized by only bilateral
transactions involving generators and loads and not
third entities such as pools or marketers;
- each agreement between generators and loads is
assumed to be an active power exchange between one
generator and one load, i.e., let with subscript k the
generic k-th transaction, PGk and PLk the active
powers sold by the generator and bought by the load,
we have PGk=PLk.
On the basis of this model, assuming nT transactions
settled in the market, we have 2nT transaction buses.
Denoting with ST the 2nT-dimensional vector composed
by the complex power injected by generators in the first
nT rows and the complex powers of load demands in the
remaining nT rows. The vector of injected powers ST,
can be decomposed into a sum of nT terms of
transactions acting simultaneously in the power system
as follows:
ST 
nT
W
(1)
k
k 1
where Wk is a 2nT-dimensional vector characterizing
the complex powers of the generic k-th transaction in
which only two elements are different from zero
corresponding to the generator and load involved in the
k-th transaction. The following example aims to clarify
the ST vector structure:
0
PG1
0












PGnT
0
0












0
0
0






0
0
0












0
0
PGk






0
0
0

 
 
ST  















0
0




 PLk  jQ Lk 






0
0


 PL1  jQ L1 








0
0
0






0
0




 PLnT  jQ LnT 
W1
Wk
WnT
As can be noted, each generator or load holds the
position as it has been filled into the vector ST.
3
Active and Reactive Power Injections
The aim of this section is to evaluate the active power
for losses and the reactive power needed by the network
and all market participants. For this purpose, we
consider an n-nodes power system represented by its
nodal equation having each line modeled with a equivalent:
(2)
IYE
where I  I1 I 2  I i  I n 
T
represents the complex
current injection vector, E  E1 E 2  E i  E n  is
the complex nodal voltage vector and Y is the nodal
admittance matrix.
T




G







L



Equation (2) can be rewritten for a system reduced to
all nodes with a non-zero injected current. Moreover, in
this representation, we split each generator with
transactions into fictitious active and reactive generators.
Distinguishing electrical variables, associated to nG
active power generators, nL load nodes, nC reactive
synchronous or static compensators, nQ reactive power
generators and the unique slack bus with subscript P, L,
C, Q and S respectively, eqn. (2) can be reformulated in
more detail as follows:
 I P   YPP YPL YPC YPQ YPS   E P 
I   Y Y Y
 
 L   LP LL LC 0 YLS   E L 
I C    YCP YCL YCC 0 YCS  E C 
(3)
  
 
0 YQQ 0  E Q 
I Q  YQP 0
 I S   YSP YSL YSC 0 YSS   E S 
We perform a different partition of eqn. (3) as follows:
 I T   YTT YTR   E T 
(4)
I   Y Y  E 
 R   RT RR   R 
where:
I T  I TP I TL T
I R  I TC I TQ I S T

YTT


Y Y 
  PP PL 
YLP YLL 

E T  E TP E TL

T
YTR

*
*
 diag E R YRT
YTT
I *T 
1


T
 
*
*
*
*
 diag E R  YRR
 YRT
YTT
YTR
E *R 
1
*
*
 diag E R YRT
YTT
diag E T  S T 
1
1

1

 
 S W  S net
where:
*
*
S W  diag E R YRT
YTT
diag E T  S T
1
 YCP YCL 
YCC 0 YCS 


YRT  YQP 0 
YRR   0 YQQ 0 
 YSP YSL 
 YSC 0 YSS 
The row and the column of the nodal admittance
matrix corresponding to a reactive power generator are
structured with only two elements that are different from
zero. This characteristic is because such a generator is
connected to the rest of the system through the unique
line connecting it to its corresponding active power
generator.
With the adopted system representation, the complex
current vector IR will be:
1
1
I R  YRT YTT
I T  YRR  YRT YTT
YTR E R
(5)
Equation (5) represents the nR-dimensional vector
(nR=nC+nQ+1) of complex currents injected by
compensators, reactive generators and the slack bus.
These currents are dependent on the transaction current
vector, IT, and ER voltages at buses where reactive
power is injected into the system.
The specified complex current vector IT can be
expressed in terms of ET voltages and ST powers injected
at same nodes as:
1
(6)
I *T  diag E T  S T
The (2nT=nG+nL)-dimensional vector, ST, representing
(8)
*
*
*
*
 diag E R  YRR
 YRT
YTT
YTR
E *R 
Y Y Y 
  PC PQ PS 
YLC 0 YLS 


Note that, the active power injected by the SR vector
represents system losses.
The next step consists in evaluating the complex
power injected by the R nodes, SR, depending on the
active power transacted by market players. For this
purpose, the following derivation is used:
S R  diag E R I *R 

E R  E TC E TQ E S

transacted powers, is composed by active powers
injected by generators in the first nG rows and complex
powers of load demands in the remaining nL rows.
The vector of complex powers injected by the nR
nodes, SR, will be composed by reactive powers injected
by the nC compensators (QC), the nQ reactive power
generators (QQ), and the complex power injected by the
slack bus (PS+QS). With these assumptions, the SR vector
will be synthetically described as follows:
(7)
S R  jQ TC jQ TQ PS  jQ S T
and

1
*
*
*
*
S net  diag ER  YRR
 YRT
YTT
YTR
1
 E 
*
R
(9)
(10)
As can be noted, the complex power SR can be split
into the terms SW and Snet The first term exhibits an
explicit dependence on the transaction vector, ST, as well
as on voltages at generator and load buses. The second
term depends on network topology and voltages at nR
nodes.
The right-hand side of eqn. (8) can be evaluated if ER,
ET and ST are known, i.e., if the operating point is
specified. This operating point can be determined
considering all scheduled transactions acting
simultaneously on the system. This hypothesis permits
us to implement a load flow calculation where
compensators are PV buses, where P=0, and transaction
generators are split obtaining PQ buses, with Q=0
(active generators), and PV buses, with P=0 (reactive
generators).
The obtained values of ER and ET enable eqns. (9) and
(10) to be easily implemented and thus to obtain the
transaction and the network components, SW and Snet.
4
S R  S W  S net 
Active and Reactive Power
Contributions
The aim of this section is to evaluate the impact, due
to each transaction, on reactive power injected by
compensators, the slack bus and generators.
In the two following subsections we give a detailed
knowledge of active and reactive contributions of the
two terms SW and Snet leading to vector SR.
a)Transaction Power Contributions SW
The ST vector appearing in eqn. (9) can be replaced by
expression (1) giving rise to power contribution SW
depending on the sum of nT injection vectors as follows:
*
*
S W  diag E R YRT
YTT
diag E T  S T 
1
1
*
*
 diag E R YRT
YTT
diag E T 
1

R
*
*1
RT YTT diag
k 1

nT
 W  
k
k 1
 diag E Y
nT
1
E T 1 Wk  
(11)
nT
S
k
W
k
W
k 1

where S kW represents the generic contribution to the
complex power, SW, due to the k-th transaction.

k 1
1
S net 
nT
 kSnet    Sknet 
nT
nT
k 1
k 1
(12)
nT
with

k
k
net
k 1

 S
nT
k
W

 S knet 
(13)
k 1
 S 
nT
k
R
k 1
From the previous equation it is possible to evaluate
the system loss component of each transaction as

k
PLoss
 real I T0S kR

k  1,, nT
(14)
where I0 is a (nR x 1) unit column vector used to sum
losses induced on all nR nodes by the k-th transaction.
Reactive powers injected by compensators, generators
and the slack bus can be obtained considering the
imaginary part of (13):
Q R  imag S R   imag S W  S net  
 nT

 Q W  Q net  imag  S kR  
 k 1

 nT k
 imag 
S W  S knet
 k 1

 Q
nT
k 1
b) Network Power Contributions Snet
The Snet complex power contribution, as can be noted by
Eqn. (10), does not exhibit an explicit dependence on
transactions. In order to share this term among
injections, we propose a practical distribution rule.
Derived network active and reactive factors are to be
considered as fixed contributions due by market
participants since they have the right to use the network.
For this reason, we adopt a practical rule aimed to share
the complex power network contribution, Snet, equally
among all market participants as follows:
nT
nT
S

k 1
S net 


nT
S

1
k 1
and S knet representing the generic contribution to the
complex power Snet due to the k-th transaction.
At this stage, eqn. (7) can be reformulated in terms of
contributions due to each transaction as:
k
W
 

 Q
 Q knet 
(15)
nT
k
R
k 1
Once the arbitrary sharing rule for the network term
Snet has been applied, the overall power can be
univocally broken down into terms depending on
injections. We can observe that, for a fixed operating
point, the unique term strictly dependent on injections is
represented by SW. Thus, this term is the only one that
can be rigorously shared among transactions.
The total amount of reactive power injected by all
generating units, due to transactions and the network,
will be:
T
T
Q TOT
 I 0 Q W  I 0 Q net 
R
(16)
 Q TOT
 Q TOT
W
net
The Q TOT
term represents the total reactive power
W
needed to exchange the power involved in the market.
Differently, the Q TOT
term is the total reactive power
net
required by the network in order to support the specified
voltage profile. This service could be considered as a
common benefit of all market participants and,
consequently, derived costs should be recovered by all
of them as network fee.
5
Test Results
The loss allocation method presented in the paper was
tested adopting a simple three bus system and the IEEE14 Bus Test System.
We assumed all generators as PV bus whereas load
buses as PQ bus. Reactive powers of loads were
obtained fixing loads power factors equal to 0.9.
Results are provided in p.u. on 100 MVA base.
For this system, we assumed the node # 1 to be the
slack bus.
Transaction data expressed in terms of active power
agreed, voltage magnitude at sending buses and reactive
power at receiving buses are reported in table 1.
Table 1
Transactions Data
Sending Bus
Bus
#
Active
Power
Agreed
[p.u.]
1
2
3
4
0.60
0.80
0.40
0.20
Trans.
Receiving Bus
Bus
#
Voltage
Magnitude
[p.u.]
#
Reactive
Power
[p.u.]
2
3
6
8
1.045
1.010
1.070
1.090
9
14
4
13
0.29
0.39
0.19
0.10
The system operating point, evaluated through the
load flow, is illustrated in Table 2.
Bus
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Table 2
Load Flow Solution
Generation
Load
Voltage Active Reactive Active Reactive
Magnitude Power Power Power Power
[p.u.]
[p.u.] [p.u.] [p.u.]
[p.u.]
1.060
1.045
1.010
0.991
1.005
1.070
1.012
1.090
0.967
0.984
1.025
1.038
0.995
0.832
0.18
0.60
0.80
--0.40
-0.20
-------
0.44
0.20
-0.34
--0.57
-0.49
-------
---0.40
----0.60
---0.20
0.80
---0.19
----0.29
---0.10
0.39
The adoption of the load flow output permitted to
implement eqn. (7) and, consequently, to obtain nodal
loss contributions and nodal charging factors by means
of eqns. (8) and (9) as reported in Table V.
Table 3
Loss Participation Factors
Transaction From Bus
To Bus
k
PLoss
#
#
#
[p.u.]
1
2
9
-0.002
2
3
14
0.040
3
6
4
0.091
4
8
13
0.049
Total
0.178
In absence of counter flows, the developed
methodology gives rise to positive loss contributions for
each transaction meaning that, with only dominant
flows, all transactions must pay for the network usage.
The test on the IEEE-14 Bus Test System gave rise to a
more complicated situation due to counter flows. In fact,
the transaction no. 1, with its negative loss contribution,
highlights the presence of counter flows. This particular
transaction has a benefic effect on system losses and all
other market participants must remunerate it.
Table 4 reports the injections of reactive power at
each reactive generating unit. In the first row the
reactive powers injected by these generators are
reported. The same results can be obtained through the
load flow calculations, as mentioned before, or adopting
eqn. (15) of the proposed methodology. By virtue of the
proposed decomposition, these reactive injections are
shared in the two terms: the reactive for transactions, QW
in the second row, and the reactive for network
requirements, Qnet in the third row. As reported in the
last column of the same table, the total reactive power
injected by all reactive generating units is equal to
=1.340 p.u., whereas assignments of total reactive
Q TOT
R
powers for transactions are equal to Q TOT
W =1.431 p.u.
and Q TOT
net =-0.090 p.u. for network requirements.
Table 4
Reactive Power Injections
BUS #
S.B.
QR
0,366
QW
Qnet
2
Generator #
3
6
8
Total
0,031
-0,418
0,842
0,519
0,080
0,240
0,146
0,584
0,380
1,430
0,286
-0,209 -0,564
0,258
0,139
-0,090
1,340
In Table 3, we report all terms Q kW directly
attributable to transactions. The sum of the elements of
each vector Q kW , reported at the end of each column,
represents the total reactive power quantity over all
reactive generators directly belonging to a transaction.
Table 5
Transaction Reactive Power Contributions
Transactions
BUS #
1
2
3
4
QW
1
2
3
6
8
(2-9)
0,026
0,077
0,047
0,087
0,161
(3-14)
0,028
0,085
0,052
0,359
0,177
(6-4)
0,025
0,074
0,045
0,036
0,034
(8-13)
0,001
0,004
0,002
0,103
0,008
0,080
0,240
0,146
0,585
0,380
k
W
0,398
0,701
0,214
0,118
1,431
Generators
S. B.
Q
For the second term, the evaluated total reactive
power needed by the network to be energized is obtained
summing the network reactive power of all reactive
injection nodes. We note that, this total reactive power,
Q TOT
net =-0.090 p.u., responds in a physically meaningful
way since its sign is in accordance with the globally
capacitive nature of the network. This term will also be
shared among transactions following the proposed rule.
The adoption of this strategy aims to divide the total
reactive power for network requirements equally among
the four transactions, obtaining the value of –0.023 p.u.
for each generic k-th transaction. More details of this
decomposition and its implications on each generator
can be found in Table 6. In this table, the first row
reports the network reactive power injected by the slack
bus and generators whereas, in the second row, the total
amount of reactive power injected by each reactive
generator is presented.
Table 6: Network Reactive Power Contributions
BUS #
Qnet/nT
Qnet
6
Generators
S.B.
1
2
3
6
0,036 -0,026 -0,071 0,032
0,286 -0,209 -0,564 0,258
k
Q net
8
0,017 -0,011
0,139 -0,090
Conclusions
In this paper a transaction assessment method for
allocating real and reactive power services to each
transaction is developed.
The system has been represented in terms of nodal
current equations giving rise to a formulation of
complex powers at the system slack bus and reactive
generators whose real part represents the active power
for losses whereas the imaginary ones is the reactive
power for transactions and network requirements.
For what system losses concerns, the methodology
demonstrates that, in presence of counter flows, is able
to discriminate transactions causing counter flows
showing negative loss contributions for them. In this
case, these particular transactions have a benefic effect
on system losses and, consequently, all other market
participants should remunerate them.
The reactive power service is shared into two parts:
one directly attributable to transactions and another one
needed to the network to be energized. This last term
responds in a physically meaningful way since its
negative sign is in accordance with the globally
capacitive nature of the network.
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