Ha2412541260

C.Prasanna Kumari, A. Hema Sekhar / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 4, July-August 2012, pp.1254-1260
Calculation of Available Transmission Capability Based on
Monte Carlo Simulation
C.Prasanna Kumari* A. Hema Sekhar** M.Tech (Ph.D)
PG Student Associate Professor
Department of EEE

Abstract -- With the further development of creation and formation of Regional
power markets, large amount of electric power Transmission Organizations (RTO), initially called
wheeling and frequent transmission transaction for by Federal Energy Regulatory Commission
is emerging, accurately determining Available (FERC) order No.2000 and further enhanced by its
Transmission Capability (ATC) and 2003 White Paper Wholesale Power Market
broadcasting the ATC information in time are Platform”, are expected to bring increased
very important to prevent and relieve efficiency of power market through improved grid
transmission congestion effectively. In practical management and increased customer access to
power markets, there are large amount of competitive power supplies. The ATC value is
uncertainties involved in Transmission required to be posted on Open Access Same-time
Reliability Margin (TRM) and Capacity Benefit Information System (OASIS) by FERC’s order
Margin (CBM) of ATC. Considering calculation 889, 2000 and White Paper to make competition
accuracy and time, a novel approach is reasonable and effective. The ATC value between
proposed to calculate ATC based on Monte two points is given as
Carlo simulation and sensitivity analysis ATC=TTC−TRM−CBM−ETC (1)
considering large amount of uncertainties
impacting the value of ATC. The steady state where TTC is Total Transfer Capability, TRM is
constraints and voltage stability and transient Transmission Reliability Margin, CBM is Capacity
stability constraints are taken into account. The Benefit Margin, and ETC is Existing Transmission
iterative solution based on PSASP by China Commitment including retail customer services
EPRI is developed. IEEE 14-bus test system is between the same two points.
used to verify the presented approach. The
results show that the model and algorithm is ATC computations should consider limits imposed
correct and effective. The research achievements on the system components such as thermal, voltage,
are undergoing to transfer to the application in and stability limits. In addition, uncertainties in
certain provincial power system in China, and load forecasts and simultaneous transfers should be
some new problems are suggested in the end of taken into account. For these reasons, uncertainties
paper. should be quantified properly to increase the
efficient use and the security of the system and to
Index Terms -- Available Transmission provide necessary information for system operation
Capability (ATC), Transmission Reliability and planning. Uncertainties are expressed by two
Margin (TRM), Capacity Benefit Margin margins, TRM and CBM, during ATC calculations.
(CBM), Monte Carlo Simulation, Sensitivity Factors such as transmission line and generating
Analysis unit outages may dramatically affect ATC if they
are not taken into consideration.
I. INTRODUCTION NERC defines TRM as “the amount of transfer
NORTH America Electric Reliability capability necessary to provide a reasonable level
Council (NERC) defines Available Transmission of assurance that the interconnected transmission
Capability (ATC) as a measure of the transfer network will be secure.” TRM accounts for the
capability remaining in the physical transmission inherent uncertainty in system conditions, its
network for further commercial activity over and associated effects on ATC calculation, and the need
above already committed uses. ATC is an for operating flexibility to ensure reliable system
indication of the expected transfer capability operation as system conditions change.
remaining on the transmission network. It is
available transfer capability that could be scheduled There are several calculating TRM approaches:
to the designated path under the conditions
considered in calculating ATC values. ATC is a Take a fixed percentage of TTC (say 4%) as TRM.
market signal that refers to the capability of a This approach is relative conservative and not
system to transport or deliver energy above that of accurate.
already subscribed transmission uses. The

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Monte Carlo statistical approach, which is based on reflects the recent market activities, while it is hard
the repeated computation of TTC using variations to deal with uncertainties. Monte Carlo simulation
in the base data, gives the difference between the in [11] can process uncertainties of power systems
maximum and the minimum. This kind of within a shorter computation time.
computation is very time consuming with the
system expansion. Considering calculation accuracy and time, a
novel method is proposed in this paper to calculate
Sensitivity analysis approach is realized by ATC based on Monte Carlo simulation and
calculating several gradient values to take sensitivity analysis considering large amount of
uncertainties into consideration. uncertainties, which have large impacts on the
NERC defines CBM as “the amount of firm value of ATC. The steady state constraints and
transfer capability preserved for Load Service voltage stability and transient stability constraints
Entities (LSEs) on the host transmission system are taken into account. The iterative solution based
where their load is located, to enable the access to on PSASP by China EPRI is developed. IEEE 14-
generation from interconnected systems to meet bus test system is used to verify the presented
generation reliability requirements.” Preservation approach. The results show that both model and
of CBM for a LSE allows that entity to reduce its algorithm are correct and effective.
installed generating capacity below what may
otherwise have been necessary without The paper is organized as follows. In Section II
interconnections to meet its generation reliability the Monte Carlo based probabilistic approach
requirements. The transmission capacity preserved dealing uncertainties in ATC calculation is
as CBM is intended to be used by LSE only in introduced. The mathematical model of ATC
times of emergency generation deficiencies. A LSE computation and Sensitivity analysis approach are
would get the benefit from CBM by sharing described in
installed capacity reserves with other parties
elsewhere in the interconnected network. Section III. The numerical examples are given
and discussed in Section IV. Last are conclusions
ATC is calculated hourly, daily or monthly in Section V.
based on market requirements. Certain factors
should be taken into account in ATC calculations II . MONTE CARLO BASED
such as the list of contingencies that would PROBABILISTIC APPROACH FOR
represent most sever disturbances, accuracy of load
forecast and distribution, unit commitment, system ATC CALCULATION
topology and configuration, and maintenance Using Monte Carlo simulation and
scheduling. System control devices such as voltage sensitivity analysis to calculate ATC has three
regulators and reactive power control devices also parts: i) construct the operation states of power
have a direct impact on ATC values. systems using Monte Carlo simulation; ii) assess
the states by power flow calculation; iii) calculate
The literature of ATC calculation can be divided the value of ATC by repetitive linear iteration
into two categories: deterministic methods and based on sensitivity analysis.
probabilistic methods. Repeated power flow (REP)
in [2], continuation power flow (CPF) in [3], and Monte Carlo simulation is a computer
optimal power flow (OPF) in [4] and [5] are based simulation method based on the probabilistic
on power flow computation. For these methods, the theory and the statistic technique. It can easily deal
steady state constraints can be easily considered but with the uncertainties. Its computation time nearly
dynamic stability constraints are difficult to be does not increase with power systems’ scale and
taken into account, and the computation time is complexity. It is very suitable for sampling for the
longer. Sensitivity analysis in [6] can provide system with large scale. In this paper we sample
results with shorter computation time. References the operation states according to the system
[7] and [8] adopt AI methods to deal with element’s probabilistic distribution using Monte
uncertainties of power systems with a shorter Carlo simulation as follows:
computer time, but with the system expansion, the
validity and effect of the algorithm is still to be For generators and transmission lines,
verified. These methods all belong to deterministic two operation states: operation or fault, are
methods. The probabilistic methods include: supposed while the corresponding probability
stochastic programming in [9] and enumeration distribution is two-point distribution.
method in [10] which consider uncertainties, but
the computation time will increase rapidly when For transformers, besides two states of
the size of system increases. Bootstrap procedure in operation and fault, there exist other states, such as
[12] computes the distribution of ATC value that transformer tapping in different positions. When
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sampling the states of transformers, two steps are constraints are taken into account. It is desired that
needed, firstly according to two-point distribution the method calculating ATC should be easy, fast
to randomly create a state, and then select a and efficient. The security margin concept, instead
tapping position. The corresponding probability of direct stability analysis, is adopted in this paper.
distribution of transformer tapping can be given Stability constrains here are divided into voltage
based on actual operation data. stability and transient stability. Checking these
margins after completing power flow calculation is
For loads, we think that every bus load is as an alternative of complicated stability
a random variable fluctuating around the forecast computation. The mathematical model and detailed
value, and can be modeled by normal distribution ATC calculation processes are described in Section
N(µ, σ2), where µ is the vector of the forecasted
load, and σ, which is estimated by the system III. ATC COMPUTATION MODEL
running experiment, is variance vector.
ATC Computation Model
After getting the current operation state, The objective of ATC calculation is point
the repeated linear iteration based on sensitivity by certain percentage (say 1%) step by step to
analysis is used to calculate ATC, which utilizes a repeat computing power flow until 15% increase of
tangent vector at an initial solution of power flow load, and making sure it is convergence of every
equations to predict the next operating point that is power flow and satisfied all constraints, which
closer to the right solution, then improve the indicates the present operation state at least with
precision by the repeat linear iteration. In this 15% voltage stability margin.
paper, the derived sensitivity formula describes the
relation between the injection power, the line B. Computation Processes
power flow and bus voltage by the first partial The ATC computation processes based on
derivative matrix. Monte Carlo and sensitivity analysis are listed as
follows.
A load factor λ is added to represent the Construct the initial state by sampling in terms of
increase of generator output and load demand. the probabilistic distribution of each equipment.
That can be presented as Compute power flow of the initial state, and
follows: PLi=PLi0+λKpli, QLi=QLi0+λKqli, evaluate the corresponding operation state. If all
0
PGi=PGi +λKpgi, where K is the multiplier to constraints are satisfied, then go on to Step 3, or
designate the rate of load or power return to Step 1.
change. If the node is fixed, then K is constant. Repeatedly increase load power in the sink and
Then the power flow equation is changed from generators’ output in the source, and compute
F(δ, V)=0 to F(δ, V, λ)=0, and the changes of λ relevant sensitivity factors as following:
correspond to the change of the operating pattern a. Compute the sensitivities of the constraints with
of power systems. respect
Using the first-order sensitivity of bus voltage, line to the load factor λ, including ∂V/∂λ, ∂Sij/∂λ,
power and generator active power with respect to λ ∂PG/∂λ, expressed by (4) to (8).
to describe the interaction of the system
parameters. That is to say: Using ∂V/∂λ the Max J  ∑PLd (2) a) Solving ∂V∂λ

sensitivity of bus voltage to λ, ∂Sij/∂λ the d R
sensitivity of line power to λ and ∂PG/∂λ the
sensitivity of generator active power to λ to The constraints include:
describe the interaction of the system parameters. N
When the system parameters changed, the load
factor λ can be decided by the first order V (G cosδ  B sinδ )  0
Pi −Vi j∑1 j ij ij ij ij
sensitivity. Then the updating ATC value can be N
obtained.
V (G sinδ − B cosδ )  0
Qi −Vi j∑1 j ij ij ij ij
The sensitivity analysis can calculate ATC
min max
sententiously, quickly and exactly, and enhance Vi ≤Vi ≤Vi i N
precision by the repeated linear iteration.
min max
Sl ≤ Sl ≤ Sl l NL (3)
min max
In ATC calculation, the thermal PGk ≤ PGk ≤ PGk k NG
constraints, voltage constraints, generator capacity
min max
constraints, load demand, and power flow balance QGk ≤ QGk ≤ QGk k NG

1256 | P a g e

−45 ≤δi ≤ 45 i N
~ * Bc * * *
& *
Where R is the set of load in sink, PLd refers to the
Sij Vi [V i ( j 2 ) (V i −V j ) y ij ] ,
load linked
we can get
to bus d; N is the number of nodes, and NL the number
min
of branches, NG the number of generators; Vi and 2
Vi
max
are the Pij Vi gij −ViV j (gij cosδij bij sin δij )
Lower and upper voltage limits of bus i, respectively;
min 2 B c
Sl
max
and Qij  −Vi ( 2 bij ) −ViV j (gij sin δij −bij cos δij )
Sl are the thermal stability constraints of branch l;
min max min max
PGk , PGk , and QGk , QGk are the output 2 2
limits of generator k. Sij  Pij Qij (6)
Δi is the phase angle of voltage of bus i. This
paper adopts the lemma “the absolute values of Then
phase angles of all buses voltages are less than 45” ∂Sij ∂Sij ∂V ∂Sij ∂V j ∂Sij ∂δ ∂Sij ∂δ j
as the simplified criterion to verify transient  i   i
stability in [6]. After every power flow calculation, ∂λ ∂Vi ∂λ ∂V j ∂λ ∂δi ∂λ  ∂δ j ∂λ (7)
checking the absolute value of phase angle of each where
bus voltage to see whether within 45, if it is, then ∂Sij ∂Pij ∂Qij
the system is regarded as transient stability.  1 (P Q )
S
For voltage stability assessment, an experiential ∂Vi ij ij ∂Vi ij ∂Vi
criterion: more than 15 percentage margin far from ∂Sij ∂Pij ∂Qij
voltage collapse, is taken instead of computing the  1 (P Q )
S ij
voltage collapse point under each state, which is ∂Vj ij ∂V j ij ∂V j
realized by increasing load in the sink From power ∂Sij ∂Pij ∂Qij
flow equations F (δ, v, λ)  0 , we can get  1 (P Q )
S
∂δi ij ij ∂δi ij ∂δi
∂P ∂Sij ∂Pij ∂Qij
∂P ∂P dδ  1 (P Q )
S
∂δ ∂V ∂λ ∂δ j ij ∂δ j ∂δ j
dV 0. ij ij
∂Q ∂Q ∂Q (4)
dλ
where Pij and Qij are real power and reactive power
∂δ ∂V ∂λ
flowed in branch ij; yij, g ij and bij are parameters
Then
of line ij; Bc is susceptance of line-to-ground.
∂δ ∂P ∂P −1 ∂P
∂ PG
∂λ  − ∂δ ∂V ∂λ (5) c) Solving
∂V ∂Q ∂Q ∂Q ∂λ
∂P
∂λ ∂δ ∂V ∂λ
G
∂P ∂P ∂P ∂λ  K Pg (8)
∂δ ∂V
where is the Jaccobian matrix; ∂∂Qλ is a column
∂Q ∂Q where KPg is the change factor of generator’s real
∂δ ∂V power. b. Compute ∆λ by (9), (10) and (11).
∂λ

0
Vector. Amongst all elements, except those KPi or V −V
lim,i i

KQi related to the sink and the source are not zero ∆λVi  (9)
∂Vi ∂λ
elements, the remainder is zero. 0
Sij P −P
b) Solving ∂ G,lim G

∂λ
∆λ G  ∂PG ∂λ (10)
From the branch power flow equation 0
− Sij
S
∂Sij ∂λ  0
lim,i , j

∂Sij ∂λ (11)
∆λ S  0
−S lim,i −Sij
∂Sij ∂λ  0
, j

∂Sij ∂λ

1257 | P a g e

Where when ∂Vi/∂λ>0, Vlim,i is the upper limit; when
0
∂Vi/∂λ<0, Vlim,i is the lower limit; Vi is the initial
magnitude of Vi; ∂Vi/∂λ is the sensitivity of Vi with
respect to λ; Slim,i j is the thermal limit of branch ij;
∂Sij/∂λ is the sensitivity of power in 76
branch ij with respect to λ; PG,lim is the maximum
Gen. 3
Gen. 8
0 74 Gen. 6
output of generators in the source point; PG is the
initial output of all generators in the source point; 72

ATC （ MW ）
∂PG/∂λ is the sensitivity of PG with respect to λ. 70
c. Find the minimum increment of ∆λ.
68

∆λ  min{∆λV , ∆λS , ∆λG } (12) 66

d. Modify λ. 64

(n1) (n)
λ λ ∆λ (13) 62
0 10 20 30 40 50 60 70 80
Generator’s output (MW)
Fig. 1. The effect of the generation on ATC

After n times of iteration, if ∆λ is less than the by China EPRI is developed. IEEE 14-bus test
system is used to verify the presented approach.
preset error ε, then to stop iterative computation, The single line diagram and parameters of the test
and the last λ is the system can be found in [15]. Bus 2 is supposed as
the source bus, bus 12, 13 and 14 are selected as
the sink buses, then ATC is defined as:
ATC=P12+P13+P14. This paper will verify the
Maximum value under the specified condition. approach from the flowing three aspects:
e. Update load and power generation by (14). (1) The influence of active power change of one
generator on ATC value. In the test system, there
0 are 5 generators: bus 1 is set as the slack bus; bus 2
PLi  PLi  λK pli
is the source bus, so consider the influences of bus
0 3, 6, 8 separately. The constraints of the generator’s
QLi  QLi  λKqli (14)
active power are: G3 (15MW, 80MW), G6
0 (10MW, 45MW) and G8 (10MW, 40MW). Select
PGi  PGi  λKPgi
5MW as the incremental active power, then active
power change of each generator is used to study the
influences on the value of ATC with the results
where PLi and QLi are real and reactive load power shown in Fig.1. Analyzing Fig.1 we can
of bus i, Kpli and Kqli are the multipliers to
designate the rate of load change, KPgi is the conclude: the active power increase of non-source
multiplier to designate the rate of power change, generator has the influence on the value of ATC;
PGi is active power generation, and superscript 0 is and different electrical distance of non-source
the initial state. generator has different influence. In the tests
(4) Compute power flow of the updated state, if generators 6 and 8 are near the source, their
power flow is convergent normally, then evaluate influences to ATC are more evident than generator
the corresponding operation state. If all constraints 3 that is far from the source bus. It also suggests
are satisfied, then return to Step 3, or go to Step 5. active power generation and unit commitment have
(5) The ATC computation is terminated if any obvious influences on the value of ATC.
of the constraints is not satisfied. The value of ATC
can be calculated by the recent power flow results (2) The influence of load change on ATC. Select
as follows. 0.02 per unit as the step, increase the load of each
ATC= ∑PLd node from 80% to 120% of the forecast value, then
(15)
d R we can get the relation curve of load change with
ATC shown in Fig.2. From Fig.2 we can conclude:
IV. NUMERICAL TESTS when the load increases, the value of ATC will
[14] decrease. It refers to the accuracy of load forecast is
The iterative solution based on PSASP/UPI
1258 | P a g e

important in ATC calculation. calculation should consider not only the influences
of single uncertainty, but also the ones of these
70
uncertainties’ combination. For example, multi-
68
fault and multi-uncertainty impact the value of
ATC at the same time. After sampling and
ATC (MW)

66
calculation, we can get a series of results
representing different states, selecting the
64 minimum one as the finial ATC value. Compare to
deterministic methods, the presented approach in
62 this paper can give the value of ATC more
accurate. For this test, we select the result of Case 5
60 as the finial result. If we take out CBM (5%), the
58
finial ATC is 57.20495%=47.3438MW.
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
Load variation (p.u.)
We are undergoing to transfer the research
Fig. 2. The influence of load changing on ATC achievements to the application in certain
provincial power system in China.
(3) The influence of different uncertainties on
ATC. Suppose the actual load value changes within V. CONCLUSIONS AND NEXT WORK
2% of forecasting value, the fault probability of According to the ATC definition of North
the generators is supposed at 0.02, the fault rate of America Electric Reliability Council (NERC), this
the lines is 0.01, and the fault rate of the paper mainly studies two margins of ATC: TRM
transformers is 0.02. Assume transformers’ tapping and CBM. Considering calculation accuracy and
have 5 levels, the probability distribution of each time, a novel method is proposed to calculate ATC
level is: 100% 0.5, 102.5% 0.15, 97.5% 0.15, 105% based on Monte Carlo simulation and sensitivity
0.10, and 95% 0.10, respectively. Using Monte analysis considering large amount of uncertainties
Carlo sampling in terms of the probabilistic involved in TRM and CBM, which have large
distribution of each equipment to construct a series impacts on the value of ATC. The steady state
of system states, and then use sensitivity analysis to constraints and voltage stability and transient
get corresponding ATC values, and select the stability constraints are taken into account. The
iterative solution based on PSASP by China EPRI
minimum as the final result. The examples are
is developed. IEEE 14-bus test system is used to
shown in Table I.
verify the presented approach. The results show
In Table I, Case 1 is the IEEE standard state, there that the model and algorithm is correct and
is no load change no fault; Case 2 only has load effective. Compare to the deterministic methods,
change; Case 3 have load and transformers’ tapping the proposed approach gives the value of ATC
adjustment simultaneously; Case 4 have load more accurate. When we transfer the theoretical
change and one line fault; Case 5 has load change achievements to practical applications, we found
and one non-source generator fault. some work needed further study.
TABLE I
ATC CALCULATION RESULTS
1. The proposed approach in this paper, like
most of other methods, is mainly suitable for
Case ATC/MW theoretically calculating ATC at a time point. But
1 63.662 in practical power markets, load and trading
2 61.357
scheme are continuously change, so it is necessary
3 65.373
for ATC computation approach to consider load
demand and relevant bidding strategies continuous
4 58.937
variation. The states, stochastically produced in
5 57.204
terms of probability distribution introduced above,
should be related to current system operation, and
when simulating continuous ATC calculation, it
Analyzing Table I, we can conclude that
should be additionally considered the transfer
uncertainties play an important role in ATC
probability from the current state to next one. Unit
calculation. For example, load change and
commitment and maintenance scheduling even
transformer tapping adjustment make the value of
bidding strategies should be taken into account.
ATC great change; the fault of line, transformer
and generator will decrease the value of ATC. Load
2. Existing Transmission Commitment
change randomly all the time, the fault of
(ETC) should be considered more detailed in
components and transformer tapping can be viewed
continuous ATC calculation. Because in power
as specific probability distribution. The ATC
1259 | P a g e

markets, ATC is an important signal to all [9] C. Y. Tsai, C. N. Lu. "Bootstrap application in
participants, and the decision and evaluation about ATC estimation," IEEE Power Engineering
the risk of different trading is also worth of Review, vol. 21, pp. 40-42, Feb. 2001.
consideration. [10] S. X. Zhou, L. Z. Zhu, X. J. Guo, Power
system voltage stability and control. Beijing:
How to improve ATC is an attractive issue. FACTS China Electric Power Press, 2004.
device is a good choice, and it can not only [11] Power system analysis software package
improve the value of ATC, but also improve the PSASP 6.1 User’s Manual. Beijing: Electric
stability of the system. But how to decide the Power Research Institute, 2001
optimal installation position of the FACTS device [12] F. Wang, G. B. Shrestha, "Allocation of TCSC
and how to coordinate multiple FACTS devices are devices to optimize total transfer capability in
still need further research. a competitive power market," in Proc 2001
IEEE PES Winter Meeting, vol. 1, pp.
VI. REFERENCES 587~593.
[1] Transmission Transfer Capability Task Force.
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[2] C. S. Wang, G. Q. Li, Y. X. Yu, W. Jiang, China, on Dec 31, 1980. She received the B.S.
"Study on transmission transfer capability of degree in Electrical Engineering, from North
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