04762620-libre

2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia
Allocation of Loss Cost by Optimal and
Proportional Tracing Methods
Nahid Aslani Amoli* and Shahram Jadid**
Center of Excellence for Power System Automation and Operation
Electrical Engineering Department, Iran University of Science and Technology (IUST), Tehran, Iran
* Email: [email protected]
** Email: [email protected]
Abstract—Allocation of transmission loss between network
users is a challenging and contentious issue in a fully
deregulated system. Also, the cost of loss must be
compensated in a fair manner by users which use the
transmission network. Power flow tracing can find the
extent of network usage by the users that can be used for
loss allocation. In this paper, proportional tracing and
optimal real power tracing methods are used for allocation
of loss and its cost with presence of congestion in
transmission lines. GAMS software is employed for solving
optimal real power tracing as linear programming problem.
The results obtained by proportional and optimal tracing
are compared in the modified IEEE 14-bus system and the
results show that optimal tracing is more fair approach than
proportional tracing.
Keywords üPower flow tracing; Loss allocation;
Proportional sharing principle; Linear programming;
I. INTRODUCTION
In recent years, electricity systems worldwide were
restructured in order to introduce market concepts. One of
the problems to be faced is the allocation of transmission
loss between network users. Also, the cost of the losses
has to be paid by the market participants which have
access to the network and the allocation must be
transparent and non discriminatory. For this purpose, it is
necessary to assess the extent of network usage by the
participants. The advent of the tracing methods solves the
problem of finding the extent of use of a network. The
proportional sharing principle has been used to develop
different methods for loss allocation. References [1-3] are
examples of these methods, where the results of a
converged power flow are used along with a linear
proportional sharing rule to allocate transmission losses
between network users. In these methods, the power flow
of generators and loads is traced to determine the
transmission system usage by each generator and load.
Then, transmission losses caused by each user are
specified. In [4], loss is attributed to a particular generator
or load in proportion to its share on the line power flows.
By this method, loss allocation can be done in exact
manner. Abhyankar et al. [5] proposed that with regard to
multiplicity of solution space in real power tracing, this
problem can be formulated as a linear constrained
optimization problem. Obtained results of solving optimal
real power tracing can be used for loss allocation between
system loads in a fair manner. The authors implemented
the proposed method for power flow tracing in MATLAB
software. Nevertheless, the allocation of loss cost between
system loads was not investigated in [5]. Since optimal
real power tracing is a nonlinear programming problem
and with regard to high ability of GAMS software in
solving such problems, in this paper, after problem
linearization, it is solved as linear programming problem
by GAMS. Also, allocation of loss cost between loads
with presence of congestion in transmission lines by
optimal and proportional tracing methods is investigated.
Due to lines congestion, optimal power flow results
instead of AC power flow results are employed by tracing
methods. This paper is organized as follows. In section II,
the concepts of proportional tracing method for loss
allocation are presented. Formulation of optimal real
power tracing and description of related constraints are
stated in section III. In next section, the obtained results of
solving optimal tracing problem by GAMS are compared
with proportional tracing results in the modified IEEE 14-
bus system. Section V concludes the paper.
II. THE CONCEPTS OF PROPORTIONAL TRACING
A. Proportional Sharing Principle
The main principle used for power flow tracing in [1-3],
is that of proportional sharing. This principle is explained
with the help Fig. 1. The figure shows bus A, where there
are two inflows of real power and two outflows. The lines
i and j carry 20 and 80 MW of power to bus A
respectively. Hence total power inflow to bus A is 100
MW. The lines p and q carry 60 and 40 MW of power
away from bus A. As electricity is indistinguishable and
the power flowing through lines is dependent only upon
voltage gradient and impedance of these lines, it may be
assumed that each of the MW leaving bus A contains the
same proportion of the inflows. Hence, 60 MW of outflow
on line p contains )
100
20
60( MW of power flowing through
line i and )
100
80
60( MW of power flowing through line j.
Figure 1. Proportional sharing principle
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2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia
Similarly, 40 MW of outflow on line q contains
)
100
20
40( MW of power flowing through line i and
)
100
80
40( MW of power flowing through line j [4].
B. Loss Allocation
1) Upstream Tracing Algorithm
Upstream tracing begins at a pure sink. Pure sink node
is defined as a node which has no outflows associated
with it. It corresponds to node with the lowest node
angle )(
δ . If one traverses the gragh in the direction of
power flow, it will not be possible to revisit the same point
(no circular flow). In the upstream algorithm, nodes are
eliminated in the ascending order of the node angles. A
node elimination operation involves:
1) Deletion of the node and its associated components
like lines, generators, loads, etc.
2) Insertion of tagged fictitious load at sending end to
maintain flow equivalence with the rest of the network.
For this purpose, proportional sharing rule is employed.
Step 1 and 2 are carried out repeatedly, till all the nodes
left to be deleted are pure sources.
2) Loss Modeling
The loss allocation is based on a simple proposition
that, as the individual entities’ flows over a transmission
line are shared in proportion to the nodal outflows, the
losses can also be shared on the same basis. Let s
lm
P and
r
lm
P be sending and receiving end powers in MW on line
lm respectively. Let virt
Li
P be the virtual load in MW of
load i, on node m, which has come out of deletion of the
earlier node. Now, the loss in the line lm, )( r
lm
s
lm PP −MW
can be allocated to load i as follows:
1
()
i
LL
i
i
virt
L
lm s r
Plmlm nvirt
L
i
P
Loss P P
P
=
=−×
¦. (1)
Where, L
n is total number of loads in the system. An
algorithm for upstream tracing after incorporating losses
can be given as below:
Step 1: Start with a pure sink node. Delete pure sink
node.
Step 2: Insert tagged virtual load i
L
Pat sending end l as
follows:
lm
ii
sr
virt virt
lm lm
LL
rr
lm lm
lm
PP
PP
PP
∀
=¦. (2)
Step 3: After each deletion of a pure sink, there is at
least one pure sink node left in the system. Delete this
pure sink node and repeat step 2.
Step 4: Repeat step 2 and 3 till all the nodes left in the
system are pure sources.
Equation (1) assigns losses over a line to various loads.
Total losses incurred due to load i
L
P are given as follows
[6]:
LL
ii
lm
PP
lm
L
oss Loss
∀
=¦. (3)
In this method, the total system loss is allocated to
loads. In the next section, loss allocation to various loads
by optimal real power tracing will be discussed.
III. OPTIMAL TRACING
A. Defining an Optimal Tracing Problem
The optimal tracing problem can be summarized as
follows:
Problem OPT(x,y):
{}
,
min
x
yS∈(, )
f
xy. (4)
The set S represents the set of all possible tracing
solutions and specific set of x and y vectors represents a
solution to generation and load tracing problem. The set S
can be characterized by a set of linear equality and
inequality constraints.
These constraints are grouped into the following
categories:
• flow specification constraints for series branches, i.e.,
transmission lines and transformers;
• source and sink specification constraints pertaining to
shunts, e.g., generators and loads;
• conservation of commodity flow constraints.
Inequality constraints are associated with flow bounds.
In the following subsection, mathematical representations
for these constraints are given.
1) Flow Specification Constraints
Traditionally, two types of tracing problems, viz.,
generation tracing and load tracing, are discussed.
Generation tracing traces generator flows to loads, while
load tracing traces load flows to generators. First,
modeling of the flow specification constraints for
generation tracing is discussed.
a) Generation tracing
Let lm P(MW) be the flow on a line lm. Flow
lm
Psupplied from system generators, is presented as
follows:
12
... nG
GGG
lm lm lm lm
PPP P=+++ . (5)
The component of generator k
Gon line lm can be
expressed as fraction k
lm
xof the total injection by
generator k
G, i.e., k
G
P
. Therefore
.
k
k
Gk
lm lm G
PxP=. (6)
1
.
G
k
n
k
lm lm G
k
PxP
=
=¦, set of lines∀. (7)
Since the branch flows are known and x are unknown,
flow equations for generation allocation can be written as
follows:
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2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia
[][][]
dd
flow flow flow
Ax b=. (8)
Matrix d
f
low
Ahas b nrows and bG nn×columns, where
b
nis the number of branches and G nthe number of
generators.
b) Load Tracing
The power flow lm Pon line lm can also be expressed as
a summation of load components, i.e.,
12
... nL
L
LL
lm lm lm lm
PPP P=+++ . (9)
The component of load ( )
i
L
P
on line lm is expressed as
a fraction i
lm
y
of load i
L
P
as follows:
.
i
i
Li
lm lm L
PyP=. (10)
Thus,
1
.
L
i
n
i
lm lm L
i
PyP
=
=¦, set of lines∀. (11)
The matrix form of above equation is:
[][][]
uu
flow flow flow
Ay b=. (12)
Matrix u
flow
Ahas b nrows and bL nn×columns,
where L
nis the number of loads.
2) Source and Sink Specification Constraints
a) Generation Tracing
In a generation tracing problem, it is necessary to write
sink (load) constraints. They express contribution of
generators in loads. This statement is presented by
following constraint:
1
G
ik
n
k
LiG
k
PxP
=
=¦. (13)
Where k
i
G
L
Pis the component of load i
L
P
met by
generator k
G. In the matrix form, the load equations for
generation allocation can be written as follows:
[][][]
dd
inj inj inj
Ax b=. (14)
Matrix d
inj
Ahas L nrows and LG nn× columns.
b) Load Tracing
In the load tracing problem, it is necessary to model the
share of loads in a generator, i.e.,
1
L
ki
n
i
GkL
i
PyP
=
=¦. (15)
In the matrix form, generator equations for generation
allocation can be written as follows:
[][][]
uu
inj inj inj
Ay b=. (16)
Matrix u
inj
Ahas G nrows and GL nn× columns.
It should be noted that all x-fractions are restricted from
0 to 1. These limits correspond to flow bound constraints.
The lower limit ensures that the flow component should
have the same direction as the arc flow, while the upper
limit ensures that no flow component exceeds the
corresponding generation. It is worth mentioning that in
traditional tracing methods [1-3], the fractions
k
lm
x,
i
lm
y
,k
i
xand i
k
y
are frozen by application of the
proportional sharing principle. In the optimal tracing,
these fractions are decision variables and are set as a result
of the optimization problem.
3) Conservation of Commodity Flow Constraints
The conservation of flow constraints can be neatly
expressed by using arc or bus incidence matrix M of the
underlying graph. In the matrix M, rows correspond to
nodes and columns to arcs. The entry ( , )
Mij is set to 1
if arc j is outgoing at node i; it is -1 if the arc is incoming
at node i; else , it is set to zero.
a) Generation Tracing
Let ( , )
ϑν ε represent the graph of network, where
ν
represents a set of all nodes and
ε
the set of arcs. Let
ε
be partitioned as follows:
{}{}{}
b
nLG
eee
ε =** . (17)
Where subset }{b
n
e represents the set of series
branches, subset
{
}
G
e indicates the set of shunt branches
due to generators, and subset
{
}
L
e represents set of shunt
branches due to loads. Then, partitioning of
ε
induces the
following column partitions on M :
[,,]
b
nLG
MMMM=. (18)
Further, let d
M represent submatrix of M formed by
considering series branches and shunt loads,
[,]
b
dnL
MMM=. (19)
Finally, conservation of commodity flow constraints for
generation tracing are stated as follows:
[]
k
dk
Mx e
ªº
ªº
=¬¼
¬¼ ,1...
G
kn=. (20)
Where k
xrepresents the set of x-variables for lines and
loads associated with the kth generator. k is the node at
which the kth generator is connected, and k
e is the kth
column of identity matrix. The set of the continuity
equations (20) can be rearranged and written in block
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2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia
matrix notations with flow xand inj xvariable partition, in
the same way as shown in (8) and (14), as follows:
flow
cont cont cont
f
low inj d
dd
inj
x
AA b
x
ªº
ªº
ªº
⋅=
«»
¬¼
«»
¬¼
¬¼ . (21)
b) Load Tracing
For the load tracing problem, the continuity equations
are given as follows:
[]
i
ui
My e
ªº ªº
=−¬¼
¬¼ , 1... L
in=. (22)
Where i is the node at which the ith load is connected
and [ , ]
b
unG
MMM=.
The set of above equations can be rearranged into the
following form:
flow inj u
uu
flow
cont cont cont
inj
y
AA b
y
ªº
ªº
ªº
⋅=
«»
¬¼
¬¼
¬¼ . (23)
B. Loss Allocation
The objective function of optimal tracing problem is
equitable distribution of losses between loads. On the
other hand, loss incurred in supplying demand i
L
Pis given
by:
1
G
Lii
i
n
i
PkLL
k
loss y P P
=
§·
=−
¨¸
¨¸
©¹
¦. (24)
By dividing (24) by i
L
P,the per unit loss for load i is
obtained as follows:
1
1
G
n
ii
pu k
k
loss y
=
§·
=−
¨¸
¨¸
©¹
¦. (25)
Therefore, the objective function ),( yxf in (4) is
defined as follows:
{}
1
,
L
n
i
pu L
i
f
xy loss k
∗
=
=−
¦. (26)
Where L k∗ is the ratio of total system loss to total
system load.
The aim of optimal tracing method is to compute the
closest traceable solution to the proportionate distribution
of transmission system losses. In other words, the loss
allocated to each load should be equal to L
k∗. In practical
power system, this condition may not get satisfied, and
hence, it is tried to adjust it as near as possible to L
k∗,
within the tracing framework.
Now, the OPT problem that was defined in (4) can be
explicitly formulated as follows:
min ),( yxf . (27)
0
0
dd
uu
A
b
x
A
b
y
ªºªº
ªº
=
«»«»
«»
¬¼
¬¼¬¼
. (28)
0≥− ki G
k
iL
i
kPxPy ,
{}
{}
L
G
ni
nk
,...,2,1
,...,2,1
∈∀
∈∀ . (29)
TT x]1...1,1[][]0...0,0[ ≤≤ . (30)
T
y]0...0,0[][ ≥. (31)
As can be seen, the optimal tracing is a nonlinear
programming problem that can be solved by GAMS.
More detailed explanations about above equations have
been presented in [5].
Due to congestion in the transmission lines, OPF
dispatch provides locational marginal price (LMP) at any
bus. Also, every load must be responsible for the system
losses he causes. Therefore, after assigning losses between
system loads by optimal and proportional tracing methods,
every load pays the charge according to Li
Pi
loss LMP×
for its allocated loss.
IV. SIMULATION RESULTS
The methods discussed for allocation of loss and its cost
are applied to modified IEEE 14-bus system in which bus
8 and line 7-8 (due to zero power flow) have been deleted.
The system is illustrated in Fig. 2. To obtain the locational
marginal prices (LMPs) as influenced by congestion, it is
assumed that the power flow limits have been decreased to
50% of the original value for both the 2-4 and 6-12 lines
and to 80% for the line 4-5. In order to remove the lines
congestion, OPF is run in the MATPOWER environment
[7]. The OPF results have been shown in the table I. Now,
for tracing power flows in the network, OPF results are
used. The L
k∗value for this system is 0.0367.
To assign losses by proportional tracing method, first,
the pure sink nodes in system are identified. Then, they
are eliminated in the ascending order of the node angles.
The elimination sequence for this system is as follows:
13-12-11-9-10-8-6-7-3-4-5-2
Based on this sequence, steps 1 to 4 are carried out. The
obtained results by this method have been presented in
table II. As mentioned before, the optimal real power
tracing for the aim of loss allocation is formulated as a
nonlinear programming problem that is solved using the
GAMS/CPLEX solver [8]. This problem essentially points
to the first norm that is converted to a linear programming
problem. In order to allocate losses between loads by
optimal tracing, OPF results are given to GAMS as input
data. After executing program, the fractions
k
lm
x,i
lm
y
,k
i
xand i
k
y
are obtained. By using (24), the loss
allocated to load i is specified. The table II shows the
obtained results. In this table, all loss allocations are
greater than zero which means that no cross subsidies on
the system unlike incremental loss allocation methods.
Indeed, the aim of optimal tracing method is to
minimize the sum of absolute deviations of per unit loss of
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2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia
all loads from the L k∗value. In other words, it aims to
achieve overall nearness of all loads to L
k∗. The objective
function value for the results obtained by optimal tracing
method is 0.048, while the sum of absolute deviations
calculated for proportional tracing results is 0.098. The
comparison of these figures shows that by optimal tracing
method, system losses are distributed between loads in a
more equitable manner rather than by proportional tracing
method. Fig. 3 depicts per unit loss allocated to various
loads, calculated by optimal as well as proportional
tracing methods. It obviously shows that the majority of
system loads near to L
k∗value by optimal tracing method
that it implies equitable distribution of losses between
system loads. This fact is not derived from the results of
proportional tracing method. The table III shows the load
payments for loss. In optimal tracing, the total load
payments is 383.63 ($ )hr , while in proportional tracing,
it is 386.47 ($ )hr .
Figure 2. The Modified IEEE 14-bus system
It is clear that a total load payment in optimal tracing is
a little less than that in proportional tracing. Of course,
this result is helpful for system loads from a fair
viewpoint, but it can be ignored due to little difference in
load payments. By changing amount or number of lines
involved in congestion, the obtained result may become
better. Overall, these results verify the effectiveness of
optimal tracing method in terms of equitable loss
allocation and also a bit less charge paid by system loads.
Figure 3. Comparing per unit loss allocation by two methods
V. CONCLUSIONS
In this paper, allocation of loss and its cost between
system loads by proportional tracing and optimal tracing
methods has been investigated. Since optimal tracing is a
nonlinear programming problem, GAMS software has
been used. Because of congestion in transmission lines,
TABLE III.
LOAD PAYMENTS FOR ALLOCATED LOSS
Load Bus No. Proportional
Tracing Optimal Tracing
2 12.36 19.25
3 123.28 115.01
4 80.07 96.85
5 10.32 11.24
6 15.22 16.57
8 49.47 49.08
9 17.18 14.29
10 6.4 5.1
11 10.96 9.18
12 26.31 20.42
13 34.9 26.64
TABLE I.
OPF RESULTS FOR THE MODIFIED IEEE-14 BUS SYSTEM
Bus
No.
Angle
(deg)
PG
(MW)
PL
(MW)
LMP
($/MWh)
1 0 196.48 - 36.909
2 -4.056 37.14 21.7 38.568
3 -9.741 34.9 94.2 40.698
4 -8.811 - 47.8 40.523
5 -7.576 - 7.6 39.958
6 -13.134 0 11.2 39.993
7 -11.94 - - 40.554
8 -13.586 - 29.5 40.576
9 -13.795 - 9 40.713
10 -13.59 - 3.5 40.488
11 -13.995 - 6.1 40.65
12 -14.046 - 13.5 40.87
13 -14.82 - 14.9 41.586
TABLE II.
MW LOSS ALLOCATION TO VARIOUS LOADS
Load Bus No. Proportional
Tracing Optimal Tracing
2 0.3205 0.4991
3 3.0291 2.826
4 1.9758 2.39
5 0.2582 0.2812
6 0.3806 0.4144
8 1.2193 1.2095
9 0.422 0.351
10 0.158 0.126
11 0.2695 0.2257
12 0.6438 0.4995
13 0.8392 0.6407
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