IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST 2009
1435
DCOPF-Based Marginal Loss Pricing With Enhanced
Power Flow Accuracy by Using Matrix Loss
Distribution
V. Sarkar, Student Member, IEEE, and S. A. Khaparde, Senior Member, IEEE
Abstract—On current days, many of the power markets are
adopting the practice of marginal loss pricing. The dc optimal
power flow (DCOPF) model is still mostly used for the purpose
of market clearing owing to its simplicity, robustness and higher
speed of convergence. Within the DCOPF framework, transmission line losses are represented as additional external loads on the
system. In the classical DCOPF model of marginal loss pricing, as
implemented in the New England market, the aggregated network
loss is distributed over the system nodes in a fixed ratio. To gain
more design flexibility in the distribution of system losses, a matrix
loss distribution (MLD) framework is developed in this paper
maintaining compatibility with financial transmission rights. In
MLD, each individual line is considered separately for the purpose
of loss distribution. The ratio in which the power loss in a certain
line is distributed can be different from that in which the power
loss in another line is distributed. The additional design flexibility
thus obtained assists in making a more accurate representation of
the power flow. The benefit of MLD is illustrated through a case
study. An energy reference independent application of MLD is
also suggested.
Index Terms—Estimation-based loss distribution, financial
transmission rights, locational marginal price, marginal loss
pricing, matrix loss distribution, power flow accuracy, revenue
adequacy, vector loss distribution.
Energy component of .
Loss component of .
Number of possible line-outage events.
Total number of network capacity constraints.
Bid price related to the th transaction bid.
Bid price related to the th load bid.
Offer price related to the th generation offer.
identity matrix.
Total number of lines in the system.
Nodal loss coefficient vector for the power loss
in the th line.
Loss distribution vector for the power loss in
the th line.
Column vector containing the transaction bid
variables.
Column vector containing the load bid
variables.
NOMENCLATURE
Branch-node sensitivity matrix for the
pre-contingency (or base) network topology.
Column vector containing the generation offer
variables.
Branch-node sensitivity matrix under the th
network contingency state.
Upper limit on
.
Upper limit on
.
Slack weight vector.
Upper limit on
.
Vector containing node voltage angles.
Net nodal injection vector (column) considering
all the energy schedules on the network.
Total number of transaction bids.
The value of
for the base case prepared
for calculating loss coefficients.
Total number of load bids.
Total number of generation offers.
Angle of
Parameter vector (column) representing the
net nodal inelastic loads in a particular energy
market.
.
nodal LMP vector.
Congestion component of .
Column vector containing the net nodal
injections caused by the physical equivalents of
active FTRs.
Manuscript received March 27, 2008; revised February 03, 2009. First
published May 05, 2009; current version published July 22, 2009. Paper no.
TPWRS-00233-2008.
The authors are with the Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India (e-mail:
vaskar@ee.iitb.ac.in; sak@ee.iitb.ac.in).
Digital Object Identifier 10.1109/TPWRS.2009.2021205
0885-8950/$25.00 © 2009 IEEE
Variable term signifying the power loss in the
th line.
Power loss occurring in the th line for the base
case prepared for calculating loss coefficients.
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Estimated value of power loss in the th line.
Variable representing the slack power.
Resistance of the th line.
Reactance of the th line.
Nodal admittance matrix.
Column vector containing the line loading
limits.
Variable vector (column) representing the line
flows in the pre-contingency network.
“From” end node of the th line.
Total number of nodes in the system.
Offset term in the linear expression of the
power loss in the th line.
“To” end node of the th line.
vector of all zeros.
vector of all ones.
Value of the scalar or vector variable
at the
solution point of the power dispatch problem.
Social-welfare function.
Optimal social-welfare as the function of
.
I. INTRODUCTION
ESTRUCTURING of the power industry has posed several challenges in front of power system engineers. Efficient management of the network congestion is one of the vital
operational issues under the restructured environment. Compared to other approaches of congestion management, the locational marginal price (LMP) mechanism has found wide acceptance throughout the world owing to its inherent efficiency in
the network capacity allocation. The LMP mechanism was first
invented by Schweppe et al. [1]. Several LMP decomposition
techniques were later proposed by other researchers [2]–[10]. A
concept of continuous LMP was proposed in [11]. Locational
marginal pricing can be carried out either on a nodal basis or on
a zonal basis [12]. There is also a proposal for a mix of nodal
pricing and zonal pricing [13].
There are two approaches of the LMP calculation, namely,
the ac optimal power flow (ACOPF) calculation and the dc optimal power flow (DCOPF) calculation. The ACOPF model is
built upon the ac power flow (ACPF) relations and, therefore,
can accurately represent the power flow. But, two major shortcomings of the ACOPF model are that 1) it is too complex requiring large execution time, and 2) it may face the non-convergence problem. The ACOPF model is currently employed in
NYISO and CAISO. However, in most the markets, the LMP
calculation is carried out by using the dc power flow (DCPF)
approximation. The DCOPF model is mostly used because of
its simplicity, robustness and higher speed of convergence.
For the DCOPF approach, the implementation of locational
marginal pricing is straightforward if line losses are not included
R
in the electricity price calculation [14]–[16]. However, owing to
the fact that the power loss in a large network may not be negligible, many markets are now moving towards marginal loss
pricing. As the dc power flow model, in itself, is lossless by definition, it is required to develop a suitable methodology to represent line losses within the DCOPF framework. To maintain
convexity of the power dispatch problem, the line losses should
be expressed as linear functions of nodal injections. Two different DCOPF models of marginal loss pricing are described in
[8] and [9]. Both of these models are perfectly suitable for financial transmission rights (FTRs), and the same simultaneous
feasibility test (SFT) methodology [14], [17] as employed in the
case of lossless LMP calculation is equally applicable to these
models. It should be noted that the FTR mechanism is basically
complementary to locational marginal pricing. The primary aim
of FTRs is to guard bilateral contracts from the volatile congestion charges. The FTR concept was first introduced by Hogan
[18]. Financial transmission rights bear a great significance in
the locational marginal pricing. Therefore, it is always desirable
that the LMP model that will be employed is suitable for the implementation of FTRs.
In the most simplistic model of [8] and [9], it is assumed
that, even for the actual resistive network, the line flows can be
approximately determined by carrying out a dc power analysis
with the given injection pattern. The model cannot produce an
energy reference (i.e., slack) independent solution. In the other
DCOPF model of [8] and [9], which is currently implemented
in the New England market, the aggregated network loss is distributed over the system nodes as additional loads by means of
a fixed distribution vector. By suitably adjusting the loss distribution vector, a good level of accuracy may be obtained in the
power flow result. The particular DCOPF model exhibits energy
reference independency. However, if the slack weight vector for
the first model is chosen to be the same as the loss distribution
vector for the second model, both the models will essentially
produce the same solution for the dispatch schedules, line flows,
LMPs and the congestion components of LMPs. This implies
that both of these two models are fundamentally similar. The
only difference between them is that the reference dependency
of the first model is converted into loss distribution vector dependency in the case of the second model. However, the relative
advantage of the second model is that there is no need to update
loss coefficients and line flow sensitivity factors each time to enhance the power flow accuracy. This is because the power flow
accuracy for this model is solely governed by the selection of the
loss distribution vector. In contrast, the power flow accuracy for
the first model is governed by the slack selection that also affects
the values of loss coefficients and line flow sensitivity factors.
For both the DCOPF models of [8] and [9], the nodal loss coefficient vector is calculated by means of an ac power flow analysis
on a base case system state.
There is also an iterative approach of marginal loss pricing
suggested in [10]. For this approach, a lossless DCOPF calculation is initially carried out. Based upon the solution of this lossless calculation, an estimation of line losses is made. In the next
and main iteration, the estimated loss in each line is distributed
as additional loads over its terminal nodes in a certain ratio. The
loss factors are calculated by means of a dc power flow analysis
SARKAR AND KHAPARDE: DCOPF-BASED MARGINAL LOSS PRICING
on the system state obtained from the base iteration. Although
the particular approach provides an additional design flexibility
to more accurately represent the power flow, it experiences some
problems with FTRs. This is because, from the point of view of
net congestion collection, each line capacity is, in effect, modified by an offset term whose value cannot be known until the dispatch problem is solved. The capacity offset that is introduced
may not be negligible. Consequently, a reasonable amount of
uncertainty may become involved in the SFT process. Moreover,
the particular model exhibits energy reference dependency. The
energy reference dependency prevails even if the ACPF-based
loss factors (as in [8]) are used instead of the above DCPF-based
loss factors. In the end, it should be mentioned that it is also possible to iteratively calculate the loss factors for [8] and [9].
In this paper, we have generalized the DCOPF model employed in the New England market by using a matrix loss distribution (MLD) methodology. In the matrix loss distribution,
a loss distribution matrix, rather than a loss distribution vector,
is used to distribute the line losses over the system nodes. Unlike the ISO-NE methodology of vector loss distribution (VLD),
it is not necessary in the MLD to maintain a fixed ratio in the
distribution of the total network loss. The particular ratio can
be made automatically adjusted according to the system connectivity and the power losses in individual transmission lines.
Looking from a different angle, in the VLD, the power loss
in each line is distributed in the same ratio, whereas, there is
no such restriction in the MLD. The additional design flexibility thus obtained assists in making a more accurate representation of the power flow. Although a similar kind of flexibility
is also offered by the estimation-based loss distribution (EBLD)
methodology proposed in [10], the particular methodology, as
mentioned earlier, is not consistent with FTRs. However, the
matrix loss distribution proposed is perfectly suitable for FTRs
as there is still a very well-defined way to make revenue adequate FTR issuance. It is, in fact, shown that the traditional
simultaneous feasibility test can ensure the revenue adequacy
of FTRs even for the MLD-based marginal loss pricing. A case
study is performed in which the MLD methodology is compared
with other methodologies.
Assigning a separate distribution vector to each line is not, in
itself, a new concept. The particular application has already been
made in New Zealand [19] and in Singapore [20], [21]. However, the particular loss distribution approach is investigated in
more depth in this paper. The salient contributions of this paper
are as follows:
• addressing the LMP decomposition and the FTR settlement under the matrix loss distribution;
• providing a general basis for the selection of the loss distribution matrix;
• suggesting an energy reference independent application of
the matrix loss distribution.
The organization of the paper is as follows: The general framework of the matrix loss distribution is discussed in Section II.
The MLD concept is numerically illustrated in Section III. The
case study is carried out in Section IV. The Case Study section also suggests a suitable basis for making selection of the
loss distribution matrix. The reference independent application
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of the MLD is described in Section V. Finally, the paper is concluded in Section VI.
II. MATRIX LOSS DISTRIBUTION MODEL
In the case of the matrix loss distribution, each individual line
is separately assigned a loss distribution vector. The loss values
that are distributed are the actual line losses, which are obtained
in the solution of the dispatch problem, rather than the estimated
losses. The loss distribution vector assigned to a particular line
can be the same as or different from the distribution vector assigned to another line. The day-ahead dispatch problem can be
compactly formulated as follows:
(1)
(2)
(3)
(4)
(5)
(6)
where
Constraint (2) is the network capacity constraint considering
both the base and contingent network topologies. Apart from
writing the line limit constraints for the current (also called as
the base) network topology, a separate set of line limit constraints are also to be written for the each possible line outage
event. This is to ensure that the flow redistribution that will automatically occur following an outage event in the actual time of
operation will not threat the system security. The matrix converts nodal injections into line flows both for the forward and reverse directions assigned to the lines. The vector is converted
into nodal injections by means of the
conversion matrix .
are the line loading limits. Constraint (3) is
The elements of
the power balance constraint. The approximated linear relationships between the power losses in different lines and the nodal
injections are enforced as Constraint (4). The vector contains
the offset terms in the linear expressions of line losses, and the
other constraints are self-explanatory.
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In the above formulation, the power loss in each line is linearized around a base case system state. This is unlike the practice followed in New Zealand and Singapore. In New Zealand
and Singapore, the power loss occurring in a line is initially expressed as a piecewise linear function of the power flow over
this line. The dispatch problem is then formulated with the assumption that the social welfare maximization process will naturally minimize line losses. However, the particular approach
makes the LMP decomposition difficult. Moreover, the dispatch
problem itself becomes very complicated for such loss modeling
if the network is large. To avoid the difficulty, the procedure of
linearization around a base case is followed in this paper. This is
in line of [8]. The standard ACPF-based loss coefficient calculation using a base case analysis is demonstrated in the appendix.
The Lagrangian function of the optimization problem (1)–(6)
can be written as
LMP difference between any two nodes is invariant of . As a
result, the congestion component of the LMP difference is also
invariant of . However, the net congestion collection (NCC)
and the net loss collection (NLC) are different for different . In
order to maintain consistency with the existing standard that the
FTR payments should be funded from the net congestion collection, the value of needs to be chosen as
(15)
For the particular , the net congestion collection of the independent system operator (ISO) can be calculated as
(7)
(16)
where , , , , and are the vectors of Lagrangian multipliers. The LMP at a node is defined as the rate of the decrease
of optimal social welfare with respect to the increase of the fixed
(or inelastic) load at that node. Therefore, from the theory of
sensitivity analysis [22]
By applying the complementary slackness condition of the KKT
rule [22], [23] for Constraint (2), the expression of NCC can be
finally written as
(17)
(8)
Similarly, the net loss collection of the ISO can be calculated as
According to the KKT necessary conditions of optimality [22],
[23]
(9)
Let us define an
vector such that
plying both sides of (9) by , we obtain
. Pre-multi(18)
(14)
and
, the net congestion collection is
As
always nonnegative. Unlike the nonnegativity of the net congestion collection, the nonnegativity of the net loss collection
cannot be rigorously proven either for the proposed model or
for the existing models. However, the value of is almost equal
for the usual choice of the slack weight vector (the
to
explanation of the slack weight vector is also provided in the appendix) where each slack weight is nonnegative. Moreover, for
the practical market condition, where the offer and bid prices
are usually nonpositive.
are all nonnegative, the elements of
This, in turn, indicates that the net loss collection is usually nonnegative.
The congestion components of the day-ahead locational marginal prices are used to calculate the hourly values of FTRs.
Similar to that as for a lossless power dispatch model, the total
target payment (TTP) to the FTR holders can be calculated as
[17], [24]
Although an infinite number of , satisfying
, is possible, it can be easily observed that the loss component of the
(19)
(10)
Therefore
(11)
Thus, a conceptual set of energy, loss and congestion components ( ,
, and , respectively) of the LMP vector can be
obtained as follows:
(12)
(13)
SARKAR AND KHAPARDE: DCOPF-BASED MARGINAL LOSS PRICING
1439
that the congestion component of an LMP also remains unaffected by the dimension reduction.
TABLE I
LINE INFORMATION
III. NUMERICAL EXAMPLE
Consider a four-node and five-line system. The line information of this system is given in Table I. Node 1 is taken as the
energy reference (i.e., slack node) and the base MVA is 100.
The generation offers, load bids, and transaction bids that are
submitted in a certain day-ahead market are shown in Table II.
In addition, there is an inelastic load of 50 MW at Node 2.
Here
TABLE II
BID INFORMATION
It should be noted that
. The vector
contains
the net nodal injections caused by the physical equivalents of
active FTRs. The active FTRs [17] at a particular hour are those
FTRs that are converted into the cash at that hour. Irrespective
of the direction of congestion, the value of an active FTR is always given by the product of its MW amount and the congestion
component of the LMP differential on its path. The value of an
inactive FTR automatically becomes zero even if there is some
congestion price difference on its path. Note that in the FTR context, a path is simply defined by a source-sink pair rather than an
alternating series of nodes and lines. Now, from (17) and (19),
, the net congestion collection
it is clear that if
will be sufficient to pay full FTR credits. Therefore, the revenue
adequacy of FTRs can be ensured by means of the standard simultaneous feasibility test [14], [17].
The dimension and size of the optimization problem (1)–(6)
can be reduced by compressing Constraints (2)–(4) into the following:
The upper half of the matrix corresponds to the forward flow
limit constraints, whereas the lower half corresponds to the reverse flow limit constraints. For the sake of simplicity, no contingency case is considered here. The power loss occurring in a
line is distributed in 3:7 ratio over its “from” end and “to” end
nodes. Therefore, the matrix appears as follows:
(20)
(21)
The locational marginal prices can then be calculated by using
the following formula:
The social welfare function can be written as
(22)
and
indicate the Lagrangian multipliers associated
Here,
with Constraints (20) and (21), respectively. As before (i.e.,
before the dimension reduction), the energy, loss, and congestion components of LMPs can be obtained as
,
, and
, respectively. The value of
that is requited to calculate the vector is obtained from (4). It
is obvious that the power dispatch and LMP solutions remain
the same before and after the dimension reduction of the dispatch problem. Moreover, the shadow prices of the capacity convalues) remain unaltered [8]. This in turn implies
straints (or
In order to calculate the loss coefficient matrix, a base case must
be prepared. Here, the base case is generated by solving the
dispatch problem neglecting line resistances. The system state
(in radian) for the particular base case is obtained as
For this base case,
can be calculated as
and
(with a certain energy reference)
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TABLE III
LMP OUTCOME
TABLE IV
GENERATION OFFERS
TABLE V
LOAD PROFILE
contingency is considered in the dispatch calculation. Node 1
is taken as the energy reference.
In order to assess the accuracy of the power flow result produced by a dispatch model, an ac load flow analysis is carried
out after the dispatch calculation is completed. The solution of
as obtained by solving the dispatch problem is to be considered as the regular injection vector in the load flow analysis.
The meaning of regular injection is explained in the appendix.
For our case study, the slack weight vector in the above load
flow analysis is chosen in such a way that the slack power becomes equally distributed over the nodes from which there is
generation participation (i.e., Nodes 1, 2, 3, 8, 11, 13, 14, 15,
18, 22, 23, and 27). The injection vector obtained after solving
. Similarly, the line
the load flow problem is indicated by
flow vector obtained after solving the power flow problem is in. It is obvious that the more close are the
dicated by
and
vectors, respectively, to
and
vectors, the more is the power flow accuracy of the dispatch
and
are the solutions of
and
,
model.
respectively, for the dispatch problem, i.e.,
(23)
(24)
The elements of are expressed in MW. Finally, the power dispatch schedules (in MW) can be calculated as follows:
The LMP outcome of this power dispatch problem is shown in
Table III.
IV. CASE STUDY
A case study is performed on the modified IEEE 30-bus
system to assess the benefit of the matrix loss distribution.
In the modified 30-bus system, the resistance of each line is
taken as the one-fifth of its reactance. The line reactances are
maintained at their original values. The base MVA is taken
as 100. The capacity of a line is taken as the lowest of 150
MW and the half of its stability limit. Table IV presents the
generation offers that are submitted in a certain day-ahead
market. In addition, there is a price-insensitive generation of 40
MW at each of Nodes 3 and 8. For the sake of simplicity, it is
assumed that there is no load or transaction bid. The fixed load
profile on the system is shown in Table V. No case of network
In the ideal situation, when the dispatch problem produces a
should be equal to
perfectly accurate power flow result,
and
should be equal to
. Therefore, the accuracy of a power dispatch model can be measured, in compact
form, by means of the following indices:
Index 1) Second norm of
.
.
Index 2) Second norm of
Index 3) Maximum flow mismatch, i.e.,
.
Index 1, in essence, indicates the Euclidian distance between the
and
in an -dimensional space. In the same
points
way, Index 2 indicates the Euclidian distance between the points
and
in an -dimensional space. A lower value
of an index indicates a higher level of accuracy.
Three different dispatch calculations are performed by using
the lossless model, the vector loss distribution, and the matrix
loss distribution. For the vector loss distribution, the loss distribution vector is chosen according to the nodal load ratios [9].
For the matrix loss distribution, the power loss in a line is distributed in 1:1 ratio over its terminal nodes. The system state
obtained by solving the lossless dispatch problem is taken as
the base case for calculating loss coefficients. The loss factors
are obtained by using the standard ACPF methodology. In the
power flow analysis for accuracy calculation, the voltage magnitude at all the nodes are held at one per unit. The nodal injection
and line flow results obtained are shown in Tables VI and VII,
respectively.
Let us calculate the values of different indices to compare the
accuracy levels of different dispatch models. The index values
obtained are shown in Table VIII. It can be seen that (compared
and
beto the lossless model) the distance between
comes significantly reduced for both the vector loss distribution
SARKAR AND KHAPARDE: DCOPF-BASED MARGINAL LOSS PRICING
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TABLE VI
NODAL INJECTION RESULTS (IN MW)
TABLE VII
LINE FLOW RESULTS (IN MW)
and the matrix loss distribution. This distance (i.e., Index 1) is
almost the same for VLD and MLD. However, for the VLD,
the distance between dispatch and ACPF solutions of line flows
(i.e., Index 2) is much higher compared to that for the MLD. The
same is the situation for the maximum flow mismatch. Therefore, it can be argued that the MLD, in overall, produces a more
accurate power flow result than does the VLD.
A dispatch calculation is also carried out by using the EBLD
methodology. As before, the power loss occurring in a line is
evenly distributed over its terminal nodes. To ensure a higher
accuracy, the loss estimation and the loss coefficient calculation
are made by using the ACPF methodology instead of the DCPF
methodology that was suggested in [10]. The values of different
accuracy indices are obtained as follows:
Notice that the accuracy level of the EBLD is comparable to that
of the MLD. However, for the EBLD, the network capacity is,
in effect, modified by an offset term from the point of view of
net congestion collection, i.e.,
(25)
where
As before,
indicates the shadow price of the th network
capacity constraint. The capacity offsets for the forward flow
TABLE VIII
INDEX VALUES
limit constraints are shown in Table IX. For a reverse flow limit
constraint, the absolute magnitude of the capacity offset is the
same as that for the corresponding forward flow limit constraint,
but the sign is the opposite.
Instead of the actual network capacity, the modified network
capacity described above should be considered in the simultaneous feasibility test of FTRs. Table IX reveals that the amount
by which a line capacity is modified may not be negligible.
However, the values of the capacity offsets remain unknown at
least until the first iteration of the dispatch calculation is over.
Therefore, a good amount of uncertainty gets involved in the
SFT process. Even though there is no uncertainty, if the capacity
offsets for the frequently congested lines are positive with respect to the directions of congestion, the issuance of FTRs may
be significantly hampered due to the lower line capacities in the
SFT. However, it is obvious from the discussion in the second
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TABLE IX
FORWARD CAPACITY OFFSETS FOR EBLD (IN MW)
the outcomes (i.e., dispatch solution, LMPs, and congestion
components of LMPs) of certain marginal loss pricing models
exhibit reference dependency. This can create dispute in the
selection of the energy reference. In order to strictly meet the
energy reference independency criteria, a modified application
of the MLD is discussed in this section. For the particular MLD
application, the loss factors are basically determined by means
of a dc power flow approximation of the system loss. However,
the central framework of the MLD remains the same.
A. Specific MLD Application
The Constraints (2) and (3) are essentially obtained by compressing the following constraints:
(26)
last paragraph of Section II that the above problems are not observed in the MLD as the whole network capacity can be used
in the SFT.
The selection of the loss distribution matrix can be made
based upon the indices discussed above. We need to consider
only Index 2 and Index 3 for that purpose. The reason behind
not considering the Index 1 is that the value of this index always
remains small if the loss factors are properly calculated. Now, as
mentioned earlier, in order to calculate the value of an index, an
ac power flow analysis should be carried out after the dispatch
problem is completed. It has been observed that the value of an
index does not show any strong dependency on the slack weight
vector to be used in the power flow analysis. Therefore, the slack
can be arbitrarily chosen to make the analysis. Here, we have assumed that the slack weight vector in the load flow analysis is
chosen in the conventional way with no negative entry. Now,
the accuracy assessment can be made by using either the actual
bids and offers, or the historical bids and offers. In the later case,
many matrices can be tried. However, in the former case, only
a limited number of trials can be made due to the time bound.
It seems best to combine both the approaches. In the first stage,
some potentially accurate matrices should be identified on a
long term basis based upon the historical data analysis. In the
next stage, the accuracy analysis should be carried out with the
actual data. The accuracy analysis for the second stage is repeated for each hour. Only those matrices that have been selected after the first stage of analysis are to be considered in the
second stage. Note that the entire process is a “trial and error”
method in which the explicit objective is to attain the lowest
possible values of the different indices.
(27)
(28)
It can be observed that the nodal injection vector
is, in effect, divided into a flow component
and a loss
component
. The flow components of nodal injections
are used to calculate the line flows (by using the dc power flow
relations), whereas the network loss is compensated by the loss
components.
Owing to the fact that the power and current flows over a line
have almost the same magnitude in per unit, the approximate
can be derived as follows:
quadratic expression of
(29)
According to the line flow modeling made
(30)
Note that all the quantities should be expressed in per unit. After
linearizing this quadratic expression around the base case state
,
can be written as
(31)
where
V. REFERENCE INDEPENDENT APPLICATION OF MLD
It is difficult to prove the energy reference independency of
the matrix loss distribution model if the standard ACPF-based
loss factors are used. The energy reference of a load flow
problem indicates the ratio or pattern by which the slack power
is injected over the system nodes. Before proceeding for the
calculation of line flow sensitivity factors and loss coefficients
on any base case system state, an energy reference must be
clearly defined. The values of the loss coefficients and line
flow sensitivity factors are always dependent upon the choice
of the energy reference. It is because of this coupling that
Eliminating
from (31), we obtain
(32)
Therefore, the loss coefficient matrix
are obtained as follows:
and the offset vector
SARKAR AND KHAPARDE: DCOPF-BASED MARGINAL LOSS PRICING
1443
is required to verify the reference independency of the congestion component of the LMP vector:
(33)
(34)
B. Energy Reference Independency
In this section, we will verify the energy reference independency of the MLD application discussed above. It is obvious that
any that satisfies (26)–(28) and (31) will also satisfy (2)–(4),
be a point that lies in the
and vice versa. Let
solution space of (26)–(28) and (31) for the slack weight vector
. Next, consider that the slack vector be changed from
to
. Any (slack dependent) network parameter corresponding
to a particular slack weight vector is indicated by the respective
weight vector in the superscript. It can be shown that the foland
, and between
lowing relationships between
and
hold well [8]:
(35)
(36)
also
Constraints (27) and (28) are satisfied by
for the slack weight vector , because none of these constraints
contains any slack-dependent parameter. Now
(37)
Therefore,
The
the
Constraint (26) is also satisfied by
for
. Next, consider the Constraint (31).
matrix can be decomposed as follows:
(38)
where
Note that both
and
are slack independent. Now
(40)
Therefore, the congestion components of locational marginal
prices also exhibit slack independency. Clearly, the particular
MLD application produces energy reference independent results
for all the practical purposes.
C. Accuracy of the Reference Independent MLD Application
Take the same system as considered in Section IV. For the
current MLD application, the values of different indices to measure the power flow accuracy are obtained as follows:
Although the value of the Index 1 is lower for the VLD compared to the MLD, the difference is still not significant. Simultaneously, from the point of view of Index 2 and Index 3, the
MLD is still producing a far better power flow result compared
to the VLD. Consequently, the accuracy level of the MLD is
maintained to be higher than that of the VLD from the overall
point of view. It can also be noticed that the accuracy levels of
the previous and current MLD applications are comparable.
VI. CONCLUSION
(39)
Therefore, it is proved that the collective constraining effect of
(26)–(28) and (31) remains unaltered after the slack weight variation. This in turn indicates that the dispatch solution produced
by (1)–(6) is independent of the choice of the energy reference
for this particular application. As the further consequence, the
locational marginal prices and the constraint shadow prices too
do not show any dependency on the energy reference. Finally, it
A matrix loss distribution methodology is developed in
this paper for the DCOPF-based marginal loss pricing with
increased power flow accuracy. While the currently employed
DCOPF model allows only a fixed-ratio distribution of the
aggregated system loss, the proposed methodology provides
the flexibility to assign a different distribution vector to each
line. As the degree of freedom in distributing line losses is
enhanced, a better representation of the power flow is possible.
Unlike the estimation-based loss distribution (that is similar to
the matrix loss distribution), the loss values that are distributed
in the MLD are the actual line losses, which are obtained in
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST 2009
the solution of the dispatch problem, rather than the estimated
losses. The main benefit of the MLD over the EBLD is that
the MLD methodology is perfectly compatible with FTRs. In
the case of the MLD, the full network capacity can be utilized
in the simultaneous feasibility test. For the EBLD, a reduced
network capacity should be, ideally, used in the SFT, but the
amount of capacity reduction required cannot be precisely
known in advance. The performance of the MLD methodology
is compared with the performances of the existing VLD and
EBLD methodologies in a case study. The result obtained from
the case study clearly illustrates the benefit of the MLD proposed. In the end, an energy reference independent application
of the MLD is suggested. In this application, the loss factors
are calculated by means of a dc power flow methodology. The
power flow accuracy of the particular reference independent
MLD application is also verified.
Here, it is assumed that the voltage magnitudes at all the nodes
are maintained at one per unit. Let, Node 1 serves as the angle
reference. That is, the voltage angle at Node 1 is set fixed to
zero. The vector can be then written as follows:
(44)
where
The first order Taylor-series expansion of (42) around the base
system state yields
APPENDIX
LOSS COEFFICIENT CALCULATION BY MEANS OF THE
STANDARD AC POWER FLOW METHODOLOGY
Here, a distributed slack power flow formulation is used for
the determination of loss coefficients. In a power flow analysis,
a desirable set of nodal injections should be initially provided.
The desirable injections are termed the regular injections in this
paper. Now, as an arbitrary set of nodal injections may not produce a valid system state (or, any solution for the system state),
some additional power should be simultaneously injected into
the system. The additional power that is injected is termed the
slack power. In the conventional lumped slack approach, the entire slack power is injected at a single node. The distributed slack
is basically a generalized model of slack power injection. In the
distributed slack approach, the slack power is not necessarily to
be injected at a single node. Instead, it can be distributed over
multiple nodes. The ratio by which the slack power is distributed
over the system nodes is decided by the slack weight vector or
. The th element of the slack weight vector indicates the fraction of slack power that will be injected at Node . The sum of
the elements of the slack weight vector is one. It should be noted
that the lumped slack is, in effect, a special instance of the distributed slack, when only one entry in the slack weight vector
is nonzero. However, in the case of the distributed slack formulation, the energy reference is represented by a slack weight
vector, whereas in the case of the lumped slack formulation, the
energy reference is represented by a particular node.
can be divided into
Thus, the net nodal injection vector
and a slack injection
, i.e.,
a regular injection
(45)
where
Similarly, the first order Taylor-series expansion of (43) yields
(46)
Combining (45) and (46), we obtain
(47)
However, for any system state, the
:
such that
vector can be chosen
(48)
Consequently, the loss coefficient matrix is obtained as
(49)
(41)
The offset vector
Therefore, according to the ac power flow relations
can be determined as follows:
(42)
(43)
where
(50)
SARKAR AND KHAPARDE: DCOPF-BASED MARGINAL LOSS PRICING
Therefore, the value of
1445
is obtained as
(51)
The value of
is given by (41) and (42) at the base case
can be calculated by using
system state. In the same way,
(43).
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V. Sarkar (S’08) received the B.E. degree in electrical engineering from Burdwan University, Bardhaman, India, in 2002 and the M.E. degree in electrical
engineering from the former Bengal Engineering College (currently Bengal Engineering and Science University), Kolkata, India, in 2004. He is currently pursuing the Ph.D. degree in the Department of Electrical Engineering at the Indian
Institute of Technology, Bombay, India.
His current research involves power system restructuring issues.
S. A. Khaparde (M’87–SM’91) received the Ph.D. degree in 1981 from the
Indian Institute of Technology, Kharagpur, India.
He is a Professor of electrical engineering at the Indian Institute of Technology, Bombay, India. He has authored several research papers and has coauthored two books. He is a member of the advisory committee of Maharastra
Electricity Regulatory Commission, India. He is on the editorial board of the
International Journal of Emerging Electric Power Systems. His current research
activities are in the areas of power system restructuring.