Application of Cubic Spline Interpolation in Estimating Market power under Deregulated Electricity Market l l l l Shafeeque Ahmed , Fini Fathima , S.Prabhakar Karthikeyan , Sarat Kumar Sahoo ,
Shriram. S.Rangarajan2,
D.P.Kothari
ISchool of Electrical Engineering, VIT University, Vellore, Tamil Nadu, India. 632014. 2 Real time Power and Intelligent Systems Laboratory, Department of Electrical Engineering, Clemson University, SC, U.S.A 3 FNAE, FNASc, Fellow-IEEE, Hon. Fellow, ISTE, Director - Research, WCE&M, Nagpur. India. 441108.
[email protected],
[email protected], 3dpk071
[email protected] Over the years, various indices have been proposed to
Abstract:
In
a
deregulated
market
environment,
the
gauge market power. In an electricity market, it is
calculation of market power is one of the top priorities of
extremely important to calculate market power not only
the bodies involved in the market. The bodies generally
during normal operating conditions but also under the
involved such as the Generation companies (GENCOs),
unexpected changes that a system may have to face.
Distribution companies (DISCOs) look towards maximizing profit whereas the RTO/ISOs look to attain a zero market power environment.
These changes generally form the various constraints and conditions
in
an
electricity
network
system.
The
transmission line flow constraint, sudden load change,
In this paper, Must Run Share and Nodal Must Run Share (NMRS) is used as an index to calculate market
line outage etc. affect the market power by a large extent. The references of this work depict the development of
power. To focus upon the fast, accurate calculations and
various indices for market power calculation over the
understanding the trends of market power under system
years. The indices such as Lerner Index (L1), Herfindahl
changes such as various
load conditions, transmission line
outage and generator outage, Cubic's Spline Interpolation is used as a technique to interpolate between market powers
Hirschman index (HHI) and Must-Run Ratio (MRR) have been used to calculate market power.
calculated over a f ew operating condition set points. A demonstration of the employment of such a system is represented in this paper which will help in achieving the objectives of the players or firms involved in the electricity market. This system will be dynamic enough to suggest the operating conditions for a desired level of market power for
The HHI is used to measure the market concentration that will reflect the number of players in the market and also the inequality in their market shares [1]. The HHI is defined as the sum of the squares of market shares of all the players as given in Equation (1).
any of the above named firms under unexpected system changes. The work is done on IEEE 14 bus test system on the MATLAB version R2012b.
N
HHI=
Must Run Share, Nodal Must Run Share (NMRS), Cubic's
(1)
i=!
Index Terms- Market power, deregulated environment, Spline Interpolation.
IS?
where N is the number of players and S is the i
/h player
market share in percentage. If the value of HHI is greater I. INTRODUCTION
T
than 1000 in percentage basis then it indicates the existence of market power. The Lerner Index measures or
he knowledge of market power lends the ability to a
seller firm, to act in a manner to comfortably raise and keep the prices of goods above the competitive levels for
indicates the proportional deviation of price at the firm's profit-maximizing output from the firm's marginal cost at that output. It is defined as shown in Equation (2).
a long period of time without any loss in the sale of
(2)
goods. The commodity in consideration is electricity which cannot be stored. Thus the concept of market
Where L1i is the Lerner index for a given firm i, Pi
power becomes more complex when applied to an electricity market [2]. The existence of market power is
and
an indication of an uncompetitive market and from the
Cid is the elasticity of demand felt by the firm. The Must
perspective of the system operators, such a phenomenon should be ceased for a healthy market. Market power information is vital from the perspective of various firms like
generating
companies
(GENCOs),
distribution
companies (DISCOs) and ISOs for their own profit making. For example, the main aim of a GENCO or a
mCi
Run Ratio (MRR) represents the capacity that must be provided by a generation company to supply a given load in a congestion zone as the percentage of the maximum available capacity of the Genco. It is defined as given in Equation (3).
seller would be to have market power and thus maintain the price above the competitive prices of other GENCOs in the market. The main aim of RTO/ISO or system
are the price and marginal cost respectively and
N)J
MRR
Pd - PI - (2:1 Pg j= N gA
2:1 Pg
operators would be to achieve a zero market power
/=
environment and would hence assist the GENCOs by assigning proper maximum generation levels in terms of MW. Also, with the advent of several market players in form of GENCOs, market power has become the focus. 978-1-4799-8641-5/15/$31.00
© 2015 IEEE
N,:;.l ],rn",
Where
- 2:1 Pg j=
j,rn",
)
(3)
I,max
PI is the active power limit (import) for the Pgj,max is the generator j active power limit
given zone,
3
in the same zone, N same zone, supplier
N
gA
The work here calculates the market power of a
is the number of generators in the
g
is the number of generators owned by
A in the zone and
P
is the total load of the same
d
zone .Commonly used market power indices to calculate
generator over a bus in terms of NMRS over certain max generating values of another generator. This helps in understanding the key concept of how the market power of a GENCO can be brought down by changing the
the market power such as Herfindahl-Hirschman index
maximum generation levels of other generators. This
(HHI), Lerner index (LI) etc. are good but both fail to
concept becomes the key principal behind this work and
clearly
it should be understood that by power system generation
reflect
the
impact
of
load
variation
and
transmission constraints on market power and hence the geographic
difference
of
market
power
is
seldom
considered. The Must run ratio which though takes into account the transmission constraints but does not clearly
restructuring in terms of varying generation levels etc. we can easily achieve the zero market power environment again even under the various unexpected occurrences in the network system. Hence to accomplish this task, in
indicate the controllability of a Genco over market price.
this
Hence the paper here calculates the Nodal Must Run
interpolate the market power curve between few points.
Share (NMRS) to calculate the market power. The indices NMRS and MRS clearly recognize the impact of load
variation
on
market
power
and
geographic
difference of market due to network constraints. Must Run Share (MRS) represent the effect of load variation and NMRS represents the geographical difference of market powers. The NMRS represents the minimum must-run capacity of a GENCO to supply a given load at a node as the percentage of total load at the same node. It combines both optimization techniques and topological analysis of load flows to determine market power. The maximum generation of generators are used as set points in the work as the must run generation of the other generators depend upon it. The Equation for NMRS is shown in Equation (4). NMRSk,l .
=
work,
Cubic's
Spline
Interpolation
is
used
to
This interpolation will enable the system operators to identify
the
new
maximum
generation
levels
for
achieving back the zero market power environment in the electricity
network
market
even
under
unexpected
occurrences when the market power gets fluctuated. Cubic's
Spline
interpolation
technique
is
used
in
determining the NMRS, which has not been used in earlier literatures on calculation of market power. It forms piecewise polynomials between few NMRS values calculated using load flows. Cubic's spline interpolation is the one of the strongest techniques available in the numerical analysis for interpolation with minimum error. It has been used previously in ATC calculation and power
amplifier
linearization
[4][5][6].
All
the
mathematical expressions are already available in the
Pgmust / Pdi k,l
literatures [4], [6]. The significance of NMRS and its i= 1,2 ... N
(4)
impact
on
FACTS
device
location
in
deregulated
electricity market is discussed in [7]. Where N is the number of buses in a power system,
Pdi
Pgf,:/I/ is the contribution of the must run generator k to Pdi The back ground calculation is the load at bus i, and
•
of NMRS is available in reference [1]. The NMRS represents the minimum capacity that must be provided by the must-run generator
k to supply a given load at
node i. The exact Equation is defined as shown in Equation (5).
NMRS.kl
=
pmust
� Pd ,
=
[M-1]. p:�ust "[M-1] p � Ik
JEN
g
lJ
The
market
power
for
the
network
system
is
calculated in terms of NMRS for the generators over all the buses under different system changes. Furthermore, a particular
generator's
market
power
over
a
bus
is
considered and is studied when the maximum generation levels for another generator are varied. This is then interpolated between few points to demonstrate a method of remodeling the GENCOs to achieve zero market power environments. The remodeling is mainly in terms of assigning new maximum generation levels to the generators present in the system. The new generating
(5)
level will be observed from the trend shown through the interpolated curves.
g;
II. METHODOLOGY
Pdi, the load at bus i; j, the bus which is directly connected to bus i through transmission lines; and
[M];
the distribution matrix which is used to show how the power supplied at a node is contributed from all the generators in a system explained in reference [I]. The work represented in this paper is from the perspective of the system operators to achieve a zero market power environment for a dynamic system. An IEEE 14 bus test system is subjected to four different system changes. Under such changes, market power generally tends to fluctuate. Thus, it is necessary to maintain a zero market power environment by finding and hence assigning the dynamic maximum generating values which are chosen as the operating points for the GENCOs.
1.
STEPS INVOLVED IN CALCULATION OF NMRS
Step I: Define the number of generators and their active power limits. Step 2: Determine
p;;us/ of generator k
Step 3: Calculate distribution matrix [M I] Step 4: Calculate NMRS of generator 1 on load l. Step 5: Repeat Step 4 for calculating NMRS for the remaining generators on each load. Step 6: Repeat step 4 and 5 for various cases. Step 7: End. •
Calculate the NMRS for all the generators using
•
Plot NMRS of a generator against the maximum
conventional method as discussed in the above steps. generation of other generators.
•
Connect two points at a time using cubic spline
NMRS and interpolating NMRS results between different
interpolation technique (piecewise polynomial) using
operating points of a generator are the main objective of
MATLAB built in function.
this paper. Thus the results obtained are categorized into five
III. SYSTEM DESCRIPTION The proposed idea of application of Cubic's spline interpolation for calculating the respective market power of generation companies on all buses for any given load 14 bus test system. The system comprises of 4 generator buses and an additional slack generator at bus number I in addition to the 9 load buses. The interconnection of the accomplished
with
20 transmission
lines.
Moreover, the transmission line flow limits are also considered
to
take
into
account
their
impacts
on
calculation of market power. When NMRS is considered as an index to measure market power in an electricity market, stress has been given to the consideration of geographical constraints and line constraints also. The single line diagram of the IEEE 14 bus data is shown in
bus), Gen2, Gen3, Gen4 and Gens are the generators located at bus I, bus 2, bus 3, bus 6 and bus 8 respectively. Before the calculation of NMRS during various operating conditions, certain assumptions are Each
generator
is
considered to have a variable working range. In order to define
the
working
range
for
an
With the help of the must run generation, the NMRS dimensional
generators,
NMRS
Genl
Gen2
Gen3
Gen4
Busl
0
0
0
0
0
Bus2
0
0.4715
0
0
0
Bus3
0
0 .171
0.2123
0.0435
0
Bus4
0
0.298
0
0.4971
0
BusS
0
0.1907
0
0
0
Bus6
0
0
0
0
0
Bus7
0
0
0
0
0
BusS
0
0
0
0
0
Bus9
0
0.0386
0
0.06
