Electrical Power and Energy Systems 31 (2009) 59–66
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Electrical Power and Energy Systems
journal homepage: www.elsevier.com/locate/ijepes
Electricity transmission pricing: Tracing based point-of-connection tariff
A.R. Abhyankar a, S.A. Khaparde b,*
a
b
Indian Institute of Technology Delhi, Department of Electrical Engineering, New Delhi 110 016, India
Indian Institute of Technology Bombay, Department of Electrical Engineering, Powai, Mumbai, Maharashtra 400 076, India
a r t i c l e
i n f o
Article history:
Received 21 November 2006
Received in revised form 16 September
2008
Accepted 18 October 2008
Keywords:
Power transmission
Point-of-connection transmission charges
Power flow tracing
a b s t r a c t
Point-of-connection (POC) scheme of transmission pricing in decentralized markets charges the participants a single rate per MW depending on their point-of-connection. Use of grossly aggregated postage
stamp rates as POC charges fails to provide appropriate price signals. The POC tariff based on distribution
of network sunk costs by employing conventional tracing assures recovery of sunk costs based on extent
of use of network by participants. However, the POC tariff by this method does not accommodate economically efficient price signals which correspond to marginal costs. On the other hand, the POC tariff,
if made proportional to marginal costs alone, fails to account for sunk costs and extent of use of network.
This paper overcomes these lacunae by combining the above stated desired objectives under the recently
proposed optimal tracing framework. Since, real power tracing problem is amenable to multiple solutions, it is formulated as linearly constrained optimization problem. By employing this methodology, consideration of extent of network use and sunk cost recovery are guaranteed, while objective function is
designed such that the spatial pattern of price signals closely follows the pattern of scaled locational marginal prices. The methodology is tested on IEEE 30 bus system, wherein average power flow pattern is
established by running various simulation states on congested and un-congested network conditions.
Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction
Transmission pricing is one of the highly discussed and debated
issues of power deregulation era. The copious literature available
on this issue substantiates this fact. Since its introduction for the
first time, the electric power market models have converged, in a
broader sense, to centralized dispatch and decentralized dispatch
concepts [1]. The choice and paradigm of transmission pricing
mechanism to be adopted largely depends on the market models
stated above.
The general principles of transmission pricing are laid down in
[2]. The centralized markets generally employ the marginal pricing
schemes [3]. Even though the marginal pricing paradigm satisfies
the important principle of providing most efficient signals for generation and transmission investments, it fails to recover the
embedded costs of transmission [4]. To recover the embedded
costs of transmission, a top-up of marginal prices with a complementary charge has been proposed in [5,6]. Another limitation
associated with marginal pricing schemes is that the charges for
the transmission systems are based on generation costs rather than
the costs of the network elements [7].
In decentralized markets, two commonly employed philosophies for transmission pricing are: point-to-point tariff and the
* Corresponding author. Tel.: +91 22 25767434.
E-mail address: [email protected] (S.A. Khaparde).
0142-0615/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijepes.2008.10.007
point-of-connection (POC) tariff [8]. The point-to-point tariff is also
called transaction based tariff which is specific to particular sale of
power from a named seller to a named buyer. A method based on
calculation of contribution of each contract on voltage angles and
consequently on power flows is presented in [9]. Various versions
of MW–Mile [10], contract path methods essentially represent the
class of transaction based embedded cost methods. However,
embedded cost based evaluation and pricing, while recovering an
agreed upon level of revenue for the utility, provides no basis for
efficient operation on the part of the network utility and no basis
for efficient spatial location and network based spatial response
for the users [11]. According to [2], most of the systems which
do not use their energy prices to send signals about the transmission, should have separate transmission prices which do send price
signals.
The non-transaction based POC tariff is employed in the Nordic
pool [12,13]. The basic principle of POC tariff is that payment at
one point, the point of connection, gives access to the whole network system, and thus the whole electricity market place. Those
entities which take part in power market activity (generators,
loads), pay a single charge in $/MW towards network usage. This
charge is decided by the connection level of that particular entity.
The POC tariff depends on the characteristics of the individual seller or buyer. The distinguishing feature of POC tariff is that it can be
applied to power exchange (PX) trades as well as bilateral transactions between two parties. As mentioned in [14], a non-transaction
60
A.R. Abhyankar, S.A. Khaparde / Electrical Power and Energy Systems 31 (2009) 59–66
based tariff should charge the user of the system with the relevant
marginal cost of transmission. A consequence of the adoption of a
non-transaction based tariff is that the geographical distance between buyers and sellers of power does not affect transmission
charges. However, as the charges differ between various points of
connection to the system, the transmission tariff may induce a generator to choose a certain location [14].
In [15,16], ex-ante transmission pricing scheme has been developed based on the real power tracing. The method determines the
share of every network user in a particular line’s average power
flow based on flow tracing and the statistics of the user’s previous
network usage. The advantage of ex-ante pricing scheme is that the
participants of the power market know the price before trading begins. Moreover, since the calculations are based on distributing
sunk costs, the methodology can be used as a financial instrument
for recovery of transmission investments. The method provides an
innovative way of determining POC tariff by assessing the network
usage of each participant in ex-ante manner.
As mentioned in [17], a transmission pricing scheme should recover the sunk cost of the transmission system in an equitable
manner, while minimizing impact on the efficiency of the shortterm markets. Also in [18] it is stated that the transmission price
system has to be defined that does not alter the market decisions,
related with the operation of the existing capacity. According to
[8], the transmission price system based on ‘sunk’ costs tends to alter the economic dispatch. Hence, it is mentioned that the transmission prices must compromise between signalling short run
marginal costs and offering reasonable assurance of cost recovery
over the long run. In [19], it is mentioned that the spatially variate
usage based access charges in two part tariff may strengthen the
price signals provided by marginal costs, if the method used to calculate access charges gives similar signals to those of marginal
charges. It is shown that there is a high similarity between marginal charges and loss allocation based on tracing. However, when
it comes to sunk cost distribution based on tracing, its pattern may
or may not match with that of loss allocation or marginal cost. This
is because, the sunk costs of transmission are of no concern while
obtaining the optimal power flow solution. In the light of the
above, the lacuna associated with the methods proposed in
[15,16] is that they tend to alter the price signals based on market
decisions. Moreover, the signals under network congestion situation do not get reflected in the ex-ante prices. This is because only
consideration given to decide the point charges is the sunk cost of
network elements.
The above discussion provides the following essential guidelines for the design of the point charges:
1. The point charges should be decided so that they recover the
sunk or embedded costs of network [2,4,7,17].
2. However, these point charges should be based on ‘extent of use’
of the network by participants [15,16,20,21].
3. The spatial variation of point charges should be in tune with
that of economically efficient price signals, to the maximum
possible extent [8,11,22].
The principle of recovery of embedded costs calls for use of
any of the methods under rolled-in pricing paradigm. The extent
of usage can be quantified by means of power flow tracing. To induce efficient use of transmission grid and the generation resources by providing efficient price signals, the spot price theory
was developed in [23]. The objectives achieved by means of efficient pricing are enlisted in [11,25]. Therefore, the third guideline
demands that the point charges should be in tune with the spatial
variation of marginal costs. The discussion on the efficiency of the
price signals provided by marginal costs is out of scope of this
paper.
Looking at the key features demanded by an ideal POC scheme,
the following options of calculating the POC tariff can be tried out
by combining conventional mechanisms under different
principles:
Use of flat rate throughout the system: This is the postage
stamp method and has obvious disadvantages as mentioned earlier. Thus, it neither provides efficient price signals, nor it
accounts for the usage of network by various entities.
Combining schemes under guidelines ‘1’ and ‘2’: This option
involves carrying out conventional proportionate tracing either
on a number of power flow simulation states or on the historical
data. The power flow tracing enables one to find out usage of
network element by each participant and its further participation in sunk cost of that element. This type of ex-ante pricing
scheme is proposed in [15,16]. Though this scheme satisfies
the first two principles stated above, nothing can be claimed
about the principle of providing economically efficient signals.
So long as loss cost allocation is considered, it is shown in [24]
that the spatial variation of marginal pricing and tracing based
pricing follows almost the same pattern. However, same may
not hold true for network element sunk cost allocation. It is well
known fact that sunk cost of each network element varies on
various parameters like interest on loan, return on equity,
debt-equity ratio, O&M charges, interest on working capital,
depreciation, etc. Thus, spatial variation of loss allocation and
sunk cost allocation may not match for a realistic system. The
results for sunk cost allocation based on conventional proportionate tracing may not follow the pattern of efficient price
signals.
Scheme based on marginal prices: In this scheme also, the
point charges are calculated using number of simulation states.
However, the difference is that the optimal power flow is carried
out with generator cost minimization as an objective function.
This enables calculation of nodal marginal prices. And then,
the sunk cost distribution is done in proportion to the marginal
prices, which subsequently decides the point charges. However,
this scheme does not accommodate the second guideline stated
above, as ‘extent of use’ of the network by participants is not
quantified. Calculation of ‘extent of usage’ is important because
it relates the sunk costs of the transmission network to the point
charges. It is worthwhile to note that the network sunk costs do
not figure in the calculation of nodal marginal prices.
It can be seen that none of the above conventional choices can
satisfy all three requirements of point charges. This paper aims at
developing a POC tariff for decentralized market participants by
accommodating all of the above mentioned guidelines. To accommodate the third condition, we make use of generalized framework
of tracing rather than conventional proportional tracing. To achieve
this, we exploit the multiplicity of solution space in real power
tracing. Most of the tracing algorithms and applications developed
so far, employ proportionate sharing principle [26–28] to choose an
answer from the large solution space. However, recently a novel
approach to tackle the large solution space of real power tracing
based on optimization technique was proposed in [29,30]. This
framework enables us to freeze to a tracing solution with an objective function that depicts the requirements. A couple of applications of the proposed approach are presented in [29,31], while
performance of the proposed approach in the presence of circular
flows is presented in [32]. In this paper, we define an objective
function for this optimal tracing problem such that the point-ofconnection charges calculated considering network sunk cost distribution employing real power tracing, try to follow spatial variation of marginal prices, as closely as possible. Thereby, it ensures
conformation to the above guidelines. It should be noted that every
A.R. Abhyankar, S.A. Khaparde / Electrical Power and Energy Systems 31 (2009) 59–66
feasible solution within the generalized framework of tracing is a
valid tracing solution. The solution based on conventional proportional tracing represents one of the feasible solutions among these.
This paper is organized as follows: The paper starts with the
concept of locational transmission price (LTP) in Section 2. In Section
3, brief account of optimal tracing framework is provided. The
objective function that satisfies goals of this paper is established
in Section 4. This section also introduces concept of pseudo-LMP
which mimics the pattern of spatial variation of locational marginal prices. Section 5 presents an algorithm for calculating the
point-of-connection charges. Results on IEEE 30 bus system are
provided in Section 6. Section 7 concludes the paper.
2. Proposed tracing based point-of-connection charges
2.1. Concept of locational transmission price (LTP)
Locational transmission price (LTPi) of a node reflects usage of
various transmission lines and elements by load or generator on
that node, which in turn is decided by the results of real power
tracing. Let TClm be the transmission service cost of line or element
lm. As mentioned earlier, this component essentially represents the
sunk costs of that element which includes the capital cost, depreciation, return on investment, operation and maintenance costs,
etc. Real power tracing makes it possible to distribute this cost
accurately among all the constituents of the power market. Let b
be the fraction in which the cost of network element is shared by
the loads. Similarly, (1 b) represents the fraction in which the
network cost is shared among the generators. In the discussion
to follow, we assume that only loads make payment. However,
similar formulation can be developed for generators.
Let ðPrlm Þ be the receiving end real power flow (MW) through the
network element lm in the simulation state s. Then,
cL;ðsÞ
lm ¼
bTC lm
r;ðsÞ
P lm
ð1Þ
be the transmission usage price per MW of the transmission elei;ðsÞ
ment lm in the state s, as applied to loads. Let ylm be the fraction
of load i on line lm obtained by tracing. Since,
r;ðsÞ
P lm ¼
nL
X
i¼1
the transmission usage cost at ith bus for load PLi , of the line lm, is
L;ðsÞ i;ðsÞ
given by ðclm ylm PLi Þ. Thus, the total transmission system usage
cost for a load i is given by
L;ðsÞ
TC i
¼ P Li
X
i;ðsÞ L;ðsÞ
ylm clm
The locational transmission price (LTPi) for a load bus in state s is
obtained by dividing the above by P Li .
LTPi;ðsÞ ¼
X
i;ðsÞ L;ðsÞ
ylm clm $=MW
i;ðsÞ
ð3Þ
8lm
If the simulation is carried out for Nt states, then the average LTPi is
given as
Nt
1 X
LTPi;ðsÞ $=MW
Nt s¼1
ð4Þ
The approaches proposed in [15,16] make use of topological generation distribution factor (TGDF) method [26], based on proportioni;ðsÞ
ate sharing principle to calculate the ylm fraction and thereby
i;ðsÞ
i,(s)
calculate LTP . Let us denote these as ylmðpÞ and LTP i;ðsÞ
p , respectively. Hence, as per Eq. (4),
LTPip ¼
Nt
1 X
LTPi;ðsÞ
p $=MW
N t s¼1
ð5Þ
Thus, LTP ip represents the point-of-connection charges based on
sunk cost distribution, with conventional tracing based on proportionate sharing principle.
2.2. Multiple solutions to LTPi
As mentioned earlier and shown in [29,30], there are multiple
i;ðsÞ
answers to ylm and the moot question is ‘which solution to use?’
By making use of optimal tracing framework presented in [29,30]
to exploit the multiplicity of solution space in real power tracing,
this paper answers the above question. Let LTP oi;ðsÞ be the locational
transmission price depicting the point-of-connection charges for
load bus i, calculated by the optimal tracing framework. Then,
LTPio ¼
Nt
1 X
LTPi;ðsÞ
o $=MW
N t s¼1
ð6Þ
The paper aims at calculating that solution of LTP io from large solution space of tracing such that least deviation of the pattern of its
spatial variation is created from the economically efficient price signals. Hence, an objective function that depicts similarity between
spatial variation of marginal prices and LTPio is developed. This is
elaborated in Section 4. In the next section, we provide the formulation of optimal tracing problem in brief. Detailed explanation and
theory behind all terms can be found in [30] and is not reproduced
here due to space constraint.
3. Defining an optimal tracing problem
As mentioned earlier, the conventional tracing algorithms [26–
28,34] employed the proportionate sharing rule to solve the problem. However, the tracing problem is amenable to multiple solutions, and to model all feasible solutions in real power tracing, in
[29,30], the problem is formulated as an optimization problem.
The optimal tracing problem presented in [29,30] is compactly
defined as follows:
Problem OPTðx; yÞ : min f ðx; yÞ
fx;yg2S
ylm PLi
ð2Þ
8lm
LTPi ¼
First step towards developing point-of-connection scheme is to
establish the average consistent power flow pattern for the given
system. This can be done either using past data or by simulating
randomly generated states around the base case. Subsequent step
involves decomposition of transmission line flows thus obtained
into the participants’ commodities. This decomposition is obtained
by real power tracing. The basic concept aims at finding the participation of each generator and load in each transmission element’s
flow and thereby decide its locational weight towards network
usage, based on the sunk costs of network elements. In the proposed scheme, the loss charges do not form part of transmission
prices. Since, POC scheme is an ex-ante scheme of transmission
pricing, designing the scheme requires simulation studies to be
done beforehand and the point charges for the future time horizon
are based on these studies. To develop the point charges, we first
introduce the concept of nodal locational transmission price (LTPi).
The LTPi takes into account the transmission sunk cost and represents the spatial distribution of network usage prices. A concept
on similar lines to that of LTP, but for energy charges was proposed
in [33] where, load pricing scheme was developed to eliminate the
merchandizing surplus.
61
ð7Þ
The vectors x and y represent the decision variables associated with
generation tracing and load tracing problems, respectively. The set S
62
A.R. Abhyankar, S.A. Khaparde / Electrical Power and Energy Systems 31 (2009) 59–66
represents the set of all possible tracing solutions and a specific set
of x and y vectors represents a solution to unified optimal tracing
problem. In [29,30], it is shown that set S can be characterized by
a set of linear equality and inequality constraints. In fact, set S is
both compact and convex. This leads to a linear constrained optimization problem. The objective function models the relationship between the flow entities and associated costs. In this paper, details
about the formulation of optimal tracing problem are not provided
due to space restriction. These details can be found in [29,30]. However, brief account of various constraints of optimal tracing problem
is given next.
3.2. Generic form of optimal tracing problem
A generic formulation of the optimization problem for real
power tracing is given as follows [30]:
min f ðx; yÞ
x
Ad 0
0
Au
bd
¼
y
bu
ð12Þ
ð13Þ
yik PLi xki PGk P 08k 2 f1; 2; ; nG g
8i 2 f1; 2; ; nL g
ð14Þ
½0; 0 0T 6 ½x 6 ½1; 1 1T
ð15Þ
3.1. Optimal tracing problem constraint modeling
½y P ½0; 0 0
For the power flow tracing problem, two versions are developed: Generation tracing and load tracing. For each version, equality constraints are grouped into following categories:
The constraints [Ad][x] = [bd] represent three classes of equality constraints for generation tracing formulation, discussed earlier. Similarly, the constraints [Au][y] = [bu] represent corresponding
constraints for load tracing formulation. Constraints (14) model
non-negativity of loss characteristics. Inequality constraints (15),
(16) model the limits on x and y variables. This generalized tracing
framework is referred to as optimal tracing problem throughout this
paper. The objective function can take various forms depending on
the application and the fairness requirements. In [29], the objective
function is proposed such that the sum of overall deviations of per
unit transmission prices of all loads is minimized, leading to a least
absolute value (LAV) problem, which further can be converted into
Linear programming (LP) problem.
The above optimal tracing formulation provides a generic
framework to solve the real power tracing problem. It provides a
modular plug and play kind of tool box which can be employed to
solve various tracing related problems with appropriate objective
function for the optimization problem, that depicts fairness. Next
section discusses the objective function that satisfies the goals of
this paper.
1. Flow specification constraints: These constraints are developed
for series branches i.e., lines and transformers. For generation
tracing problem, these constraints mean nothing but expressing
xklm as component of total injection of generator Gk on line lm.
This set of constraints is given as follows:
Plm ¼
nG
X
xklm :PGk 8 set of lines
ð8Þ
k¼1
Similar constraints for load tracing problem are developed as
follows:
Plm ¼
nL
X
yilm :PLi 8 set of lines
ð9Þ
i¼1
2. Source and sink specification constraints: These constraints
pertain to shunts, e.g., generators and loads. For generation
tracing problem, these constraints express contribution of generators in loads. This set of constraints can be represented as
follows:
P Li ¼
nG
X
xki PGk for i ¼ 1; . . . ; nL
ð10Þ
k¼1
Similar constraints for load tracing problem are developed as
follows:
P Gk ¼
nL
X
yik PLi
for k ¼ 1; . . . ; nG
T
ð16Þ
4. Objective function
Let LMPi,(s) denote the short run marginal cost or the locational
marginal price of node i, in simulation state s. As mentioned in Section 2, the objective function of Eq. (12) should be such that the
obtained by tracing results
spatial variation pattern of LTP i;ðsÞ
o
should match as closely as possible, with that of LMPi,(s).
Let total transmission cost of the network to be paid by loads be
TCL. This is known a priori and is given as
TC L ¼
ð11Þ
X
ð17Þ
bTC lm
8lm
i¼1
3. Conservation of commodity flow constraints: The conservation of commodity flow enforces that in a multi-commodity
network with zero storage capacity of nodes, the conservation
of flow constraint also holds for all commodity flow
components.
Inequality constraints are associated with flow bounds. Two approaches to model the losses in the formulation are proposed in
[29]. Since the generation tracing and load tracing problems represent two mutually exclusive problems, there is a need to bring
cohesiveness between the results of the two. This is achieved by
proposing additional inequality constraints that represent the
non-negativity of the losses.
In conventional tracing algorithms, the fractions xklm , xki of generation tracing and yklm and yik of load tracing are frozen by application of proportionate sharing principle. In the proposed approach,
these fractions are decision variables and are set as a result of optimization problem.
Let, pLMPi,(s) represent the pseudo-LMP of load bus i in state s, such
that,
nL
X
pLMPi;ðsÞ PLi ¼ TC L
ð18Þ
i¼1
The pLMPi,(s) indicates the scaling down of LMPi,(s) by factor of c.
Therefore,
nL
X
cLMPi;ðsÞ PLi ¼ TC L
ð19Þ
i¼1
Hence,
pLMPi;ðsÞ ¼ cLMPi;ðsÞ
ð20Þ
i,(s)
Multiplying original LMP
by the factor c ensures that the pattern
of spatial variation of original LMPi,(s) is maintained.
As per the guidelines provided in Section 1, the desired result of
the optimal tracing demands that in an ideal case, the LTP iðsÞ
o
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A.R. Abhyankar, S.A. Khaparde / Electrical Power and Energy Systems 31 (2009) 59–66
obtained should match with pLMPi(s) for all load buses. Hence, the
objective function of Eq. (12) can be written as follows:
min
nL
X
jpLMPi;ðsÞ LTPi;ðsÞ
o j
ð21Þ
i¼1
where, LTP i;ðsÞ
is given as:
o
LTPi;ðsÞ
¼
o
X
i;ðsÞ
L;ðsÞ
ylm;ðoÞ clm $=MW
ð22Þ
8lm
The nearness of the solution to that of pattern of pseudo-LMP can be
measured through various vector p-norms [35]. Here, l1 norm (k.k1)
is used because it leads to a least absolute value (LAV) optimization
problem which can be easily translated into a linear programming
(LP) problem [36].
Remark 1. LTP io of Eq. (23) represents the desired point-ofconnection transmission charge for ith load.
Remark 2. LTP ip shows the results calculated as per method presented in [16]. Here, it is calculated for comparison purpose only.
Remark 3. The algorithm and associated equations above are
developed for loads only. However, same procedure is followed if
generators also share the payment.
5. Implementation
As mentioned earlier, in the absence of past data, a set of different operating states needs to be simulated [15]. In this work, the
results are carried out on IEEE 30 bus system. While determining
the POC charges, all possible operating states of the power system,
including the network constrained states should be considered.
Hence, following two sets of simulations are carried out as follows:
1. 500 states are generated randomly around base case loads using
a Gaussian distribution with a standard deviation of 30% for
active and reactive power production. The network constraints
are relaxed.
2. 500 states are generated in the same manner as above, however, network constraints are enforced by random selection of
one of the four lines for reduction in power carrying capacity.
This establishes average power flows for two gross scenarios:
un-congested case and congested case. Then, depending on the history of congestion occurrence, appropriate weights can be assigned
to these two cases to establish the overall POC charges.
Optimal power flow (OPF) is carried out in each state, with generation cost minimization as an objective function so as to decide
P Gk and LMPi,(s). On these OPF results, optimal real power tracing
with the objective function of (21) and constraints (13)–(16) is carried out. Let, LTP io;c LTP io;uc represent the average locational transmission price calculated by optimal tracing under congested and
un-congested cases, respectively. Then, final LTP io representing
POC tariff is calculated as follows:
LTPio ¼ a1 LTP io;c þ a2 LTPio;uc
4. Calculate pLMPi,(s) using Eqs. (19) and (20).
5. Solve optimal tracing problem presented in Section 3 with an
objective function of Eq. (21). Also, solve tracing problem by
proportional sharing principle.
and LTP iðsÞ
for all i.
6. Calculate LTPiðsÞ
o
p
i;ðsÞ
.
iter
iter + 1.
7. Store LTP o and LTPi;ðsÞ
o
8. If iter < 500, Go to step 2.
9. Calculate LTPio and LTP ip using Eqs. (6) and (5), respectively.
Next section presents the results obtained on IEEE 30 bus system under various test conditions.
6. Results
The results are obtained on three sets of payment sharing
schemes among generators and loads. It is mentioned in [37] that
the generators’ share in transmission price in European countries
ranges from 0% to 50%. Based on this information, we carry out results on following three cases: b = 1, b = 0.75 and b = 0.5.
6.1. Results with b = 1
This is the case when only loads make payment towards transmission pricing. Fig. 1 shows the plot of LTPio that represents the
POC charges calculated by optimal tracing. For these results, the
values a1 = 0.2 and a2 = 0.8 are assumed. Same figure also plots
the pseudo-LMP for all load buses. In an ideal case, the point-ofconnection charges calculated by optimal tracing for all buses
should come out to be equal to corresponding pseudo-LMP. But,
this may not happen so, as the tracing results depend on the sunk
costs of the network elements and have a constrained solution
space. The same figure also shows LTPip for comparison. The results
show that by virtue of objective function of (21), the overall nearness of LTPio to that of pseudo-LMP is more than that of LTPip .
ð23Þ
i
The factors a1, a2 represent the weights given to the congested and
un-congested cases, respectively.
Since, data about TClm is not available, it is made proportional to
the nodal LMP difference across the line, as follows:
m
TC lm / jLMP LMP jP lm
LTPo
i
pLMP
i
LTPp
4
ð24Þ
where, Plm represents the real power flow (MW) for standard data
and LMPl, LMPm represent nodal locational marginal prices across
the line. Algorithm of Section 5.1 represents a generic step-by-step
methodology to calculate the point-of-connection charges. By using
the same algorithm, both LTP io;c and LTPio;uc can be calculated.
5.1. Algorithm to calculate LTPi for load buses
1. Initialize iter
0.
2. Create a state (s) by randomly sampling a load condition.
3. Carry out OPF for this state (s). The result of OPF will give generation dispatches, power flows and locational marginal prices
(LMPi,(s)) for all i.
$/MW
l
5
3
2
1
0
5
10
15
20
25
Bus No.
Fig. 1. The POC charges with b = 1; a1 = 0.2; a2 = 0.8.
30
64
A.R. Abhyankar, S.A. Khaparde / Electrical Power and Energy Systems 31 (2009) 59–66
Table 1
Mean and standard deviation of dio and dip for all cases.
i
LTPo,uc
i
pLMPuc
i
LTPp,uc
5
$/MW
4
3
2
dio
dip
b
a1
a2
l
r
l
r
1
0
1
0.2
0
1
0
0.8
1
36.8072
23.6121
34.1681
44.8831
41.5132
22.7633
37.76322
42.8281
44.3792
32.9127
42.0859
55.1711
49.7151
29.0573
45.58354
51.9840
0.75
1
0.2
0
0
0.8
1
34.5716
42.8208
45.0080
32.3932
40.74112
42.8383
49.6391
54.0647
55.0156
40.2358
41.5872
51.9693
0.5
1
0.2
0
0.8
33.5358
42.7135
33.5496
40.9805
50.0550
54.0234
45.3009
50.6356
1
Similarly, for conventional proportional tracing,
0
5
10
15
20
25
30
dip ¼
jLTP ip pLMPi j
pLMPi
Bus No.
Fig. 2. Plots of LTPio;uc , pLMPiuc and LTPip;uc with b = 1, a1 = 0, a2 = 1.
Fig. 2 shows the plots of LTP io;uc , pLMPiuc and LTP ip;uc . This is similar to the case when a1 = 0 and a2 = 1. Similarly, Fig. 3 shows the
plots of LTP io;c , pLMP ic and LTP ip;c . This is similar to the case when
a1 = 1 and a2 = 0.
It should be noted that the following will hold true always:
nL
X
LTPio PLi ¼
i¼1
nL
X
LTPip PLi
ð25Þ
i¼1
In other words, increase or decrease in LTP io of a particular bus i will
be at the cost of corresponding decrease or increase in LTP io of other
buses.
To find out the nearness of the LTPio with the pLMPi, the cluster
analysis for the deviation of LTPio and LTPip from that of pseudo-LMP
is done. Let dio denote the deviation of LTP io from that of pseudoLMP. Then,
jLTPio pLMPi j
5
pLMPi
100%
ð26Þ
ð27Þ
Table 1 shows the mean and standard deviations of dio and dip for
various values of b, a1 and a2.
It can be seen from Fig. 2 that under un-congested case, the
pseudo-LMPs for all buses are restricted within a tight band. However, sufficient distortion can be witnessed in case of congested
network situation (Fig. 3). It is important to note that whatever
may be the network operating state and whatever may be the
set of sunk costs of transmission network, the mean associated
with dio will always be lesser than the mean associated with dip .
This is apparent from Table 1. This means that the spatial pattern
of point charges obtained by optimal tracing will be more close to
that of marginal prices, than that obtained by proportional
tracing.
Another important point to be noted is that the tracing results are topology dependent even in the case of optimal tracing
framework. However, they are set within the boundaries of constrained region so as to meet the objective. For example, load on
bus 2 makes very less use of the network (line 1–2) and thus
has least POC charge obtained by both, optimal and proportional
tracing. On the other hand, load on bus 30 makes maximum use
of network and has highest POC charges obtained by both the
methods. However, LTPio will be adjusted (within tracing constraints) such that overall deviation from pseudo-LMP will be
minimum.
4
i
LTPo,c
i
pLMPc
i
LTPp,c
i
3
$/MW
dio ¼
100%
4
2
LTPo
i
pLMP
i
LTPp
0
3
5
10
15
20
25
30
Bus No.
1.2
i
2
LTPo
i
pLMP
i
LTPp
1
$/MW
$/MW
1
1
0.8
0.6
0.4
0.2
0
5
10
15
20
25
Bus No.
Fig. 3. Plots of LTP io;c , pLMPic and LTP ip;c with b = 1, a1 = 1, a2 = 0.
30
0
5
10
15
20
25
Bus No.
Fig. 4. The POC charges with b = 0.75; a1 = 0.2; a2 = 0.8.
30
A.R. Abhyankar, S.A. Khaparde / Electrical Power and Energy Systems 31 (2009) 59–66
Table 2
POC charges for generators and loads on same buses.
Bus No.
POC charges ($/MW)
b = 0.75
2
5
8
b = 0.5
Load
Gen
Load
Gen
0.1735
0.7210
2.1172
0.5035
0.0825
0.0228
0.1146
0.4806
1.3930
1.0212
0.1601
0.0000
POC Charges for Loads
2.5
$/MW
2
1.5
i
LTPo
i
pLMP
i
LTP
1
0.5
0
5
10
15
Bus No.
20
25
30
POC Charges for Generators
i
LTPo
i
pLMP
i
LTP
$/MW
2
1.5
1
65
means. To accomplish the same, the inherent multiplicity of solution space in tracing is utilized, rather than employing conventional proportional sharing based tracing. Recently proposed
optimal tracing framework is used for the said purpose.
The paper floats concept of locational transmission price (LTP).
Various system conditions are simulated to establish the average
power flow pattern of the system. The LTP is calculated based on
the participation of each node in each transmission line power
flow. Thus, depending on the sunk costs of that line, the LTP provides spatial variation of the network usage charges across the system. Since the real power tracing problem is amenable to multiple
solutions, the objective function is devised in such a way that the
solution obtained creates least distortion from the pattern of economically efficient price signals. Thus, the LTP at a bus represents
the POC charge at that bus.
It is seen that the results obtained by both, i.e., optimal tracing
and conventional proportional tracing, provide feasible solution to
POC charges under tracing regime. However, the POC charges provided by optimal tracing are able to match the price signals provided by marginal costs, as they are directly linked to marginal
costs through objective function. On the other hand, POC charges
by conventional tracing show indifference to the price signals.
The rigor and complexity of the scheme can be justified because
the exercise is not time critical. The simulation studies are done
beforehand and POC charges for future time horizon are based on
these studies. The POC tariff thus obtained can be employed to
charge the transaction based and non-transaction based trades.
0.5
0
References
5
10
15
Bus No.
20
25
30
Fig. 5. The POC charges with b = 0.5; a1 = 0.2; a2 = 0.8.
6.2. Results with b = 0.75
In this case, the loads are charged 75% of the network costs
while the generators are charged 25%. The results with a1 = 0.2
and a2 = 0.8 are shown in Fig. 4.
Bus numbers 2,5 and 8 have both generators and loads on it.
Table 2 enlists the POC charges for both generators and loads on
these buses, for b = 0.75 and b = 0.5.
6.3. Results with b = 0.5
The POC charge allocation when both generators and loads
share half of the cost is shown in Fig. 5.
From Table 2, it may be noted that the POC charges are in tune
with the degree of deficiency or adequacy of power on that bus. For
example, on bus 2, with b = 0.75, average generation is 97 MW,
while average load is 21.7 MW. Hence, the loads are provided
incentive by charging them lesser than the generators. Exactly
opposite phenomenon can be observed on bus 8. For the same case,
the average generation on this bus is 5 MW, while average load
amounts to 30 MW. Hence, in this case, generators are given incentive by charging them less as compared to the loads. Bus 5 represents a case in between these two extreme cases.
7. Conclusion
This paper provides a real power tracing based methodology to
determine the ex-ante point-of-connection (POC) rates for the participants of decentralized market. The desired qualities of POC
charges include consideration of network usage, recovery of sunk
costs and provision of price signals, all of which are hard to achieve
in a single scheme, if the point charges are devised by conventional
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