A MULTIPURPOSE MATRICES METHODOLOGY FOR TRANSMISSION USAGE, LOSS AND RELIABILITY MARGIN ALLOCATION IN RESTRUCTURED ENVIRONMENT

Electrical & Computer Engineering: An International Journal (ECIJ) Volume 2, Number 3, September 2013
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A MULTIPURPOSE MATRICES METHODOLOGY FOR
TRANSMISSION USAGE, LOSS AND RELIABILITY
MARGIN ALLOCATION IN RESTRUCTURED
ENVIRONMENT
Baseem Khan, Ganga Agnihotri, Gaurav Gupta
Department of Electrical Engineering, MANIT, Bhopal, India
ABSTRACT
In the era of power system restructuring there is a need of simplified method which provides a complete
allocation of usage, transmission losses and transmission reliability margin. In this paper, authors presents
a combined multipurpose matrices methodology for Transmission usage, transmission loss and
transmission reliability margin allocation. Proposed methodology is simple and easy to implement on large
power system. A modified Kirchhoff matrix is used for allocation purpose. A sample 6 bus system is used to
demonstrate the feasibility of proposed methodology.
KEYWORDS
Modified Kirchhoff Matrix, Transmission usage, Transmission Loss, Transmission Reliability Margin.
NOMENCLATURE
pf୧୨= Elements of Power Flow Matrix
p୧୨= Active Power in Branch i − j from Bus i to Bus j
p୘୧= Net Flows on the Nodes
K୫= Modified Kirchhoff Matrix
k୧୨
୫
= Elements of Modified Kirchhoff Matrix
kl୧୨= Elements of Kirchhoff Loss Matrix
p୧୨
୪
= Transmission Loss in Line i-j in Actual Direction
p୨୧
୪
= Transmission Loss in Line i-j in Counter Direction
I= Identity Matrix
P୐= Active Load Power
Pୋ= Active Generation
Pୋୋ= Number of Generators in Diagonal Matrix
SFM= Supply Factor Matrix
t୧୨= Elements of Supply Factor Matrix
P୐୐= Number of Loads in Diagonal Matrix
EFM= Extraction Factor Matrix
p ୱୠ
୪
= Loss in s-b Line
r୨ୱ= Elements of Extraction Factor Matrix
P୧→ୱିୠ
୪
= Share of Generator i in the Transmission Loss of Line s-b
pୱିୠ= Power Loss in Transmission Line s-b

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1. INTRODUCTION
In deregulated electricity market, issues like usage allocation, loss allocation, transmission
pricing, congestion management etc. are of great importance. Several methodologies have been
proposed for the allocation of usage, loss and transmission reliability margin (TRM). But every
methodology has its own limitation. Some of them (e.g., postage stamp, contract path, MW-Mile)
are based on the actual network usage of a transaction and are addressed as embedded methods,
while others (marginal/incremental) are based on the additional transmission cost that is caused
by a specific electricity transaction [27]. In a restructured market structure seller and buyer made
no direct transaction. Hence the usage allocation between them is made by power flow tracing
methods, used to calculate the contribution of each user (generator or load) to each line flow.
Colombia, UK and Brazil, have used long run marginal cost (LRMC) methodology due to its easy
implementation. Selection of the slack bus greatly influenced the pricing methodologies, such as
use of a fixed “slack bus” is adequate in countries where most of the load is concentrated in a
single centre, such as the cities Buenos Aires (Argentina) and Santiago (Chile). Hence the
marginal participation method is applied in countries like Argentina, Chile and Panama.
In India the method which is used so far is Postage stamp method. But this method is not distance
and direction sensitive. It is mainly depending on the amount of transacted power. The main
advantage of this method is its simplicity in implementation. Due to its various demerits central
electricity regulatory commissions (CERC) of India proposed a new transmission pricing
methodology which is the combination of power flow tracing technique and marginal
participation methodology. In the proposed hybrid methodology power flow tracing is used for
selection of the “slack bus” while by marginal participation burden of transmission charges or
losses on each node is computed.
Due to deregulation, the capacity margins of transmission lines is reduced because the number of
participates on electric grid is increased. Hence grid ability to transfer power from generation to
load within permissible voltage and frequency limits is also greatly decreased. So the
transmission embedded cost allocation methodologies should consider all these factors [15]. For
this purpose a factor related to transmission reliability margin should be added in transmission
pricing mechanism to address issues related to transmission network reliability.
There are various transmission pricing methodologies, which are used across the world for
allocation of transmission charges to users. These are mainly classified into the embedded cost,
and market based pricing methodologies. Embedded Cost Pricing methods are based upon
determining a utility’s total cost of providing the transmission services. It includes service, asset,
and operation & maintenance costs. While the market based pricing methodologies are driven by
a competitive bidding process which results in prices that are influenced by the demand of
services. Power flow tracing methods of cost allocation are comes under the embedded cost
pricing. These provide us a complete view of the usage allocation problem which is very
important for transmission cost allocation. When usage allocation is known it is straightforward
to allocate the transmission cost to generators and loads. The first attempt to trace power flows
was done by Bialek et al. when topological generation distribution factors based power flow
tracing were proposed in March 1996 [1], which explained the method for tracing generator's
output. In Feb 1997, Kirschen et al. [2] explained a power flow tracing method based on the
proportional sharing assumption which introduces the concept of domains, Commons, and links.
In Nov 2000, Gubina et al. [3] described the method to determine the generators’ contribution to a
particular load by using the nodal generation distribution factors. In Aug 2000, Wu et al. [4]
explained the use of graph theory to calculate the contributions of individual generators and loads
to line flows and the real power transfer between distinctive generators and loads. In 2009, Xie et

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al.[5] proposed power flow tracing algorithm founded in the extended incidence matrix
considering loop flows. In Feb 2007, Conejo et al. [6] explained a method of network cost
allocation based on Z-bus matrix.
Further, the existing loss allocation methods may be classified into prorata, marginal, power flow
tracing, and circuit theory based methods. Prorata method is characterized by the allocation of
electric losses proportionally to the power delivered by each generator and each load. It is also
assumed an equal allocation 50% to generator and 50% of the loads [7]. In marginal procedure,
incremental transmission coefficients are used for allocation of transmission losses to demands
and generators [8]. The use of power flow tracing methods for allocation of transmission losses is
proposed in [9]. In circuit theory based method, A. J. Conejo et al. proposed a Z- bus matrix for
transmission loss allocation. This method presents a new procedure for allocating transmission
losses to generators and loads in the context of pools operated under a single marginal price
derived from a merit-order approach [10]. The main difficulty in allocating losses to load or
generator to bilateral contracts by circuit theory is that, despite approximations, the final
allocations always contain a certain degree of arbitrariness. Recently several new algorithms and
methods have been also proposed such as in [11] a method based on complex power flow tracing
is proposed. In [12] author decomposed transmission losses into three components and
characterizes them. A method based on a combination of cooperative game theory and circuit
theory is presented in [13].
Due to deregulation, the number of interaction entitles on the electric grid increases dramatically.
At the same time capacity margin reduced. Hence this rapid growth threatened reliability of
transmission network greatly. That’s why transmission reliability margin allocation should be
addressed in a fair way. Many researchers proposed methodology which incorporate reliability
cost element in total transmission cost. For this first usage and TRM allocation is done. In [14]
Silva et al. considered the transmission network operation under normal as well as contingency
condition for allocation reliability cost to users. In [15-16] D. Hur et al. proposed various variants
of procedures to allocate reliability contribution to market participants. In [17] V. Vijay et al.
proposed a novel probabilistic transmission pricing methodology with consideration to
transmission reliability margin. In 2008 H. Monsef et al. [18] presented the transmission cost
allocation based on use of reliability margin under contingency condition. For this purpose a
probability index is defined.
In this paper, authors presents a combined multipurpose matrices methodology for Transmission
usage, transmission loss and transmission reliability margin allocation.
2. PROPOSED MATRICES METHODOLOGY
Let consider a simple diagraph G showed in fig. 1 [21].

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Fig. 1: A Simple Diagraph G
The Kirchhoff matrix of above diagraph is given by Eq. 1.
KሺGሻ = ൦
1
−1
0
0

0
2
−1
−1

−1
0
2
−1

0
−1
−1
2
൪
Hence from the above example for a simple digraph of vertices, an by matrix called the
Kirchhoff matrix KሺGሻ or K = [k୧୨] is defined as [21],
K = ൜
dିሺv୧ሻ for i = j
−x୧୨ for i ≠ j
(1)
Where dିሺv୧ሻ = in-degree of the ith vertex
−x୧୨

= (i, j) th entry in the adjacency matrix
This matrix is the basis of the proposed methodology.
Firstly, authors construct a power flow matrix from the Newton Raphson load flow. This matrix
gives a complete overview of power flows in the system. It is formed between nodes of the
system. Diagonal elements give net flows at nodes and off diagonal elements give the actual
flows and counter flows in the system. The proposed matrix is defined as follows:
pf୧୨ = ቐ
−p୧୨ for i ≠ j and p୧୨ > 0
p୧୨ for i ≠ j and p୨୧ > 0
p୘୧ for i = j
(2)
Where
p୧୨ ሺ> 0ሻ= active power in branch i– j from bus to bus
From the above matrix and using eq.1 the Modified Kirchhoff matrix is constructed as follows:
Denoting Modified Kirchhoff matrix of a Power Network as K୫ = ሺk୧୨
୫

ሻ୬×୬, the authors define
the following expression for elements of the Modified Kirchhoff matrix:

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k୧୨
୫
= ൝
−p୧୨ for i ≠ j and p୧୨ > 0
p୘୧ for i = j
0 otherwise
(3)
Now from the above Modified Kirchhoff matrix, Kirchhoff loss matrix can be formed as follows:
kl୧୨ = ቐ
p୧୨
୪
for i ≠ j and p୧୨ > p୨୧and p୨୧ < 0 < p୧୨
p୨୧
୪
for i ≠ j and p୨୧ > p୧୨and p୧୨ < 0 < p୨୧
0 otherwise
(4)
Where
p୧୨
୪
= p୧୨ + p୨୧, and p୨୧
୪
= p୨୧ + p୧୨
2.1. Properties of Modified Kirchhoff matrix
Property.1: The sum of all elements in the row j of a Modified Kirchhoff matrix equals the active
load power at bus j i.e [5].
K୫

I = P୐ ሺ5ሻ
Property.2: The sum of all elements in the column j of a Modified Kirchhoff matrix equals the
total active power of generators at bus j i.e.
I୘K୫ = ሺPୋሻ୘ (6)
The above equation can be rewritten as follows
K୫
୘
I = Pୋ (7)
From equations (5) and (7) we have
I = K୫
ିଵ
P୐ (8)
I = ሺK୫
୘
ሻିଵ
Pୋ (9)
eq. (9) can be rewritten as
I = ሺK୫
ିଵ
ሻ୘
Pୋ (10)
From the above matrix, inverse of Modified Kirchhoff matrix (K୫
ିଵ
) is obtained which is used
for power flow tracing and transmission loss allocation. In the next section procedure of power
flow tracing and transmission loss allocation is described.
3. PROCEDURE FOR TRACING POWER FLOW AND LOSS ALLOCATION
In this paper authors adopt the tracing procedure which is proposed in [5]. But authors modified
this tracing algorithm for transmission loss and reliability margin allocation.

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3.1 Model for Power flow Tracing
Let ln = 1........e represents the total number of lines in the system. Gn = 1........g is the total
number of generators and D = 1. . . . . . d is the total number of loads in the system.
Again let Pୋୋ = diag ሺPୋଵ, Pୋଶ, … . , Pୋ୥ ሻ represents the number of generators in diagonal matrix.
Thus
I୘
Pୋୋ = ሺPୋሻ୘
or Pୋ = PୋୋI (11)
combining eqs. (11) and (8)
Pୋ = PୋୋK୫
ିଵ
P୐ (12)
Matrix PୋୋK୫
ିଵ
is named supply factor matrix. The supply factor matrix is denoted by SFM =
൫t୧୨൯, i. e.,
SFM = PୋୋK୫
ିଵ
(13)
and from eq. (9)
Pୋ୧ = ∑ t୧୨
୬
୨ୀଵ P୐୨ (14)
Where t୧୨P୐୨ denotes the active power distribution of generation output at bus i to the load situated
at bus j [5].
P୧→୨ = t୧୨P୐୨ (15)
Thus eq. (15) gives the generator’s share to loads in the system.
On the same line for calculating the generators shares to lines flow eq. (15) is modified by
replacing load power from the lines flow as shown in eq. (16). It is assumed that a a୥ୣ୬: a୪୭ୟୢ
(23:77) split in the transmission usage occurs between generators and demand [19].
For the generator share situated at bus s to the line s-b is given by
P୧→ୱିୠ = t୧ୱPୱୠa୥ୣ୬ (16)
Eqs. (15) and (16) gives the generators share in loads and lines flows. Similarly, the usage
allocated to a load for the use of all lines can be defined by using a୪୭ୟୢ instead of a୥ୣ୬.
For calculating the loads shares in line flows and generated power same procedure is followed:
Considering dual of eq. (9)
P୐ = P୐୐ሺK୫
ିଵ
ሻ୘
Pୋ (17)
Where the diagonal matrix P୐୐ = diag ሺP୐ଵ, P୐ଶ, … . . , P୐ୢሻ and EFM= P୐୐ሺK୫
ିଵ
ሻ୘
is the
extraction factor matrix of loads from generators [5].
By using an EFM, loads share in generating power and line flows is calculated.

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3.2 Model for Transmission Loss
For transmission loss allocation to generator considers eq. (16). In this equation line flows Pୱୠ is
replaced by the transmission Loss in lines which is coming from the elements of the Kirchhoff
loss matrix p୧୨
୪
and p୨୧
୪
.
Hence transmission losses of line s-b allocated to generator located at bus i is given by:
P୧ିୱ→ୠ
୪
= t୧ୱp ୱୠ
୪
(18)
Similarly transmission losses of line s-b allocated to load situated at bus j is given by:
P୨→ୱିୠ
୪
= r୨ୱp ୱୠ
୪
(19)
From the equations (18) and (19) losses are allocated to generators and loads respectively. This
method of loss allocation is said to be direct because all the calculation is already done for usage
allocation.
3.3 Model for Transmission Reliability Allocation
For transmission reliability margin allocation to generator and load, considered eq. 16,
TRM = maximum capacity of the line in p.u. – usage of the line in p.u.
TRMij = 1-pfij (20)
Where for a particular line the calculation of TRM has considered Maximum capacity of the all
line is 1 p.u
In this equation line flows Pୱୠ is replaced by the transmission reliability margins in lines which is
coming from the elements of TRMij.
Hence transmission reliability margin of line s-b allocated to generator located at bus i is given
by:
TRM୧ିୱ→ୠ
୪
= t୧ୱtrmୱୠ
୪
(21)
Similarly transmission reliability margin of line s-b allocated to load situated at bus j is given by:
TRM୨→ୱିୠ
୪
= r୨ୱtrm ୱୠ
୪
(22)
From the equations (21) and (22) TRM are allocated to generators and loads respectively.
4. RESULT AND DISCUSSION
The proposed matrices methodology is applied to the sample 6 bus system presented in [26] bus
power system to demonstrate the feasibility and effectiveness of the methodology. A computer
program coded in MATLAB is developed.

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4.1 Transmission Usage Allocation
The sample 6 bus power system is used to illustrate the proposed methodology. Table 1 gives the
generators contributions to line flows. These tables also provide the transmission charge
allocation to generators.
Line Flow(pu)
Supplied by
Gen.1(pu)
Supplied by
Gen.2(pu)
Supplied by
Gen.3(pu)
1-2 0.29 0.29 0.00 0.00
1-4 0.44 0.44 0.00 0.00
1-5 0.36 0.36 0.00 0.00
2-3 0.03 0.01 0.02 0.00
2-4 0.33 0.12 0.21 0.00
2-5 0.16 0.06 0.10 0.00
2-6 0.26 0.10 0.17 0.00
3-5 0.19 0.00 0.01 0.18
3-6 0.44 0.01 0.01 0.42
4-5 0.04 0.03 0.01 0.00
5-6 0.02 0.01 0.00 0.00
Table.1: Transferred Power Allocated to Generators for 6 Bus System
Similarly, the Extraction factor matrix (EFM) is formulated for calculating the power extracted by
the loads from the generator bus and line flows.
Table 2 provides the transmission line flows allocated to loads.
Line Flow
Extracted
by Load4
(pu)
Extracted
byLoad5
(pu)
Extracted
byLoad6
(pu)
1-2 0.29 0.14 0.12 0.03
1-4 0.44 0.21 0.18 0.05
1-5 0.36 0.17 0.14 0.04
2-3 0.03 0.01 0.01 0.01
2-4 0.33 0.13 0.08 0.12
2-5 0.16 0.06 0.04 0.06
2-6 0.26 0.11 0.06 0.10
3-5 0.19 0.00 0.06 0.14
3-6 0.44 0.00 0.13 0.31
4-5 0.04 0.04 0.00 0.00
5-6 0.02 0.00 0.02 0.00
Table.2: Extracted Power Allocated to Loads for 6 Bus System

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4.2 Transmission Loss Allocation
Table 3 gives a transmission loss allocation to loads and generators. Total system losses occurred
in the system is 0.0847 pu from which 23% is allocated to generators and 77% is allocated to
demands.
Lines Loss L4 L5 L6 G1 G2 G3
1-2 0.0094 0.0035 0.0029 0.0008 0.0022 0.0000 0.0000
1-4 0.0113 0.0042 0.0035 0.0009 0.0026 0.0000 0.0000
1-5 0.0112 0.0042 0.0035 0.0009 0.0026 0.0000 0.0000
2-3 0.0004 0.0001 0.0001 0.0001 0.0000 0.0001 0.0000
2-4 0.0164 0.0051 0.0029 0.0047 0.0014 0.0024 0.0000
2-5 0.0056 0.0017 0.0010 0.0016 0.0005 0.0008 0.0000
2-6 0.0062 0.0019 0.0011 0.0018 0.0005 0.0009 0.0000
3-5 0.0123 0.0000 0.0028 0.0066 0.0001 0.0001 0.0027
3-6 0.0108 0.0000 0.0025 0.0058 0.0000 0.0001 0.0024
4-5 0.0004 0.0003 0.0000 0.0000 0.0001 0.0000 0.0000
5-6 0.0006 0.0000 0.0005 0.0000 0.0001 0.0000 0.0000
Total 0.0847 0.0211 0.0208 0.0233 0.0100 0.0044 0.0051
Table.3: Transmission Loss Allocation
4.3 Transmission Reliability Margin Allocation
From the equations (21) and (22) TRM are allocated to generators and loads respectively. From
the table 4 it is observed that the generators which contribute more power to line flows, have
more TRM allocated.
Line TRM G1 G2 G3
1-2 0.608 0.608 0 0
1-4 0.420 0.42 0 0
1-5 0.617 0.617 0 0
2-3 0.956 0.3537 0.61184 0
2-4 0.596 0.22052 0.38144 0
2-5 0.803 0.2967 0.51392 0
2-6 0.683 0.2527 0.43712 0
3-5 0.747 0.01494 0.02241 0.70965
3-6 0.488 0.00976 0.01464 0.4636
4-5 0.922 0.70072 0.26738 0
5-6 0.907 0.57141 0.14512 0.23582
Table.4: Transmission Loss Allocation
5. CONCLUSIONS
This paper presents a combined methodology for transmission usage, loss and reliability margin
allocation. A simple Kirchhoff matrix is used for this purpose. Various authors provide many
methods for addressing such issues but none of them addresses all these issues collectively. Also

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all the calculation is done only single time for all these allocation. Proposed method can be
applied to a large power system with FACTS devices. A sample 6 bus system is used to
demonstrate the feasibility of proposed methodology.
ACKNOWLEDGEMENTS
The authors would like to thank everyone, just everyone!
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Authors
Baseem Khan was born in Gwalior, India in 1987. He received BE degree (2008) from
Maharana Pratap College of Technology Gwalior and received an M.Tech. degree (2010)
in Power System from MANIT Bhopal. At the moment he is a research scholar at
MANIT Bhopal, India.
Ganga Agnihotri received BE degree in Electrical engineering from MACT, Bhopa l
(1972), the ME degree (1974) and PhD degree (1989) from University of Roorkee, India.
Since 1976 she is with Maulana Azad College of Technology, Bhopal in various
positions. Currently she is professor. Her research interest includes Power System
Analysis, Power System Optimization and Distribution Operation.
Gaurav Gupta was born in chhatarpur, India in 1985. He has received degree in 2006
from MITS Gwalior and received ME degree2009 in power electronics from SGSITS
indore. At present he is a research scholar in MANIT Bhopal, India.

A MULTIPURPOSE MATRICES METHODOLOGY FOR TRANSMISSION USAGE, LOSS AND RELIABILITY MARGIN ALLOCATION IN RESTRUCTURED ENVIRONMENT

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A MULTIPURPOSE MATRICES METHODOLOGY FOR TRANSMISSION USAGE, LOSS AND RELIABILITY MARGIN ALLOCATION IN RESTRUCTURED ENVIRONMENT