This paper presents a probabilistic method to evaluate the total transfer capability (TTC) by considering the sequential quadratic programming and the uncertainty of weather conditions. After the initial TTC is calculated by sequential quadratic programming (SQP), the transient stability is checked by time simulation. Also because power systems are exposed to a variety of weather conditions the outage probability is increased due to the weather condition. The probabilistic approach is necessary to evaluate the TTC, and the Monte Carlo Simulation (MCS) is used to accomplish the probabilistic calculation of TTC by considering the various weather conditions.
1. Introduction
Total transfer capability (TTC) is the largest quantity of electric power that can be transferred over the interconnected transmission networks in a reliable manner while meeting all of the pre and postcontingency system conditions
[1]
. The relationship of the TTC and available transfer capability (ATC) is described in the North American Electric Reliability Council (NERC) definition
[1]
.
At the present time, there are two techniques of methods for calculating the TTC, deterministic and probabilistic. The deterministic approaches mainly use the methods such as security constrained optimal power flow (SCOPF), continuation power flow (CPF), linear programming (LP) and repeated power flow (RPF), and have some difficulties in handling the uncertainties of power systems such as the possibility of faults and weather prediction
[2

4]
. LP is one of fast methods to search the solution for the initial TTC. In order to reduce linearization errors, load flows should be performed periodically. Probabilistic methods have considered the uncertainties of the system performance that could not be addressed in a deterministic way, and have been implemented to evaluate the TTC for various outages
[5

7]
. As all power system networks and the system components are exposed to nature, they are affected by the weather condition considerably, and the failure rates of transmission lines are increased due to weather conditions. Therefore, the operation of power system could be addressed in a probabilistic approach
[8]
.
This paper presents a probabilistic method to evaluate the TTC by considering the uncertainty of weather conditions. In the TTC evaluation, unlike the previous study of the author
[8]
, optimization method such as SQP is used to calculate the initial TTC. The weather conditions are divided into normal and adverse weather. Because the failure rate in adverse weather condition is considerably larger than that in normal weather, the contingency in power system influences TTC assessment.
2. Determination of TTC Using Deterministic Approach
 2.1 Problem formulation
In order to determine the TTC by deterministic method, the mathematical formulation for the TTC evaluation can be expressed as follows:
subject to
where,
λ_{l} : incremental factor of load & generation in outage l
P_{Gi} = P_{Gi0} (1 + λ_{l}k_{Gi}) : real power generation at bus i
P_{Li} = P_{Li0} (1 + λ_{l}k_{Li}) : real power demand at bus i
Q_{Li} = Q_{Li0} (1 + λ_{l}k_{Li}) : reactive power demand at bus i
Q_{Gi} : reactive power generation at bus i
P_{Gi0} : original real power generation at bus i
P_{Li0}, Q_{Li0} : original real/reactive power load at bus i
k_{Gi}, k_{Li} : constants specifying the rate of change in generation and load
V : voltage magnitude at bus
V_{min} , V_{max} : lower and upper limits of voltage magnitude at bus i
S_{ij} : apparent power flow in line ij
S_{ij}_{max} : thermal limit of line ij
δ_{Gi} (t), δ_{Gj} (t) : rotor angles of generator i, j
δ_{G} _{max} : maximum secure relative swing angle
For calculating the TTC, the injection real and reactive power at source and sink buses are functions of the incremental factor of load and generation in outage
l
,
λ_{l}
. The objective is to maximize incremental factor of load and generation in (1). The active and reactive power balance equations can be shown in (2) and (3). Inequality constraints such as voltage magnitude of buses and thermal limits can be written in (4) and (5). Eq. (6) means the difference between the critical energy and the transient energy as transient stability constraint. The optimization method enables transfers by increasing the complex load with uniform power factor at every load in the sink areas and increasing the injected real power at the generation buses in the source area in incremental steps until limits are incurred.
The process of transient stability analysis in Eq. (6) is very time consuming. Therefore the processes to calculate the deterministic TTC divide into two steps as shown in
Fig. 1
. The first step is to calculate the initial TTC level TTC
_{l0}
satisfying voltage and thermal limits by using optimization such sequential quadratic programming (SQP). The next step is to perform transient stability analysis based on the initial TTC level TTC
_{l0}
calculated in the first step. If there is no limit violation on transient stability, then each TTC level TTC
_{l}
is determined. If there is unstable in the transient stability, the incremental factor is decreased and the RPF is performed until the transient stability is stable. The deterministic TTC is determined by considering all factors that influence in transfer capability with above
[9]
.
Deterministic TTC calculation.
 2.2 Sequential quadratic programming
Sequential quadratic programming (SQP) is the optimization method for the minimization of the maximum of a set of smooth objective functions subject to equality and inequality constraints and simple bounds on the variables
[7]
. In order to get the optimal solutions the SQP generates a point satisfying these constraints by solving a strictly convex quadratic program (QP) using a positive definite estimate H of the Lagrangian. And an Armijotype arc search or line search (monotone, nonmonotone) are used to compute the direction of descent the objective function. Generalized the SQP algorithms are implemented as follows:

Step 1Initializationi) Initial value of variablesx0, step sizet0and search directionsd0. Ifx0is infeasible for some constraint, substitute a feasible point.

Step 2Computation of searchi) Computethe solution of the strictly convex QPii) Compute the step size

Step 3Updatesi) Update Hessian matrix of Lagrangian using the Powell modification.ii) Setxk+1= xk+tkdk+iii) Solve the unconstrained QP problem inμ, eq. (13). Increasekby 1.
where the
and
λ_{k , j}
l are the KT multipliers associated with QP for the objective functions, variable bounds, equality constraints, and inequality constraints respectively.
3. Determination of TTC using Probabilistic Approach
 3.1 Weather model with uncertainty
Because power systems are exposed to various weather conditions the failure rate of outdoor components can be increased very significantly during adverse weather periods such as gales, lighting storms, etc. Usually the components that receive the most effects of the weather are transmission lines in the power system. The transmission line can be defined to be in two states that are influenced by weather conditions, normal and adverse weather conditions
[12]
.
 3.2 Probabilistic TTC using monte carlo simulation
The sequential Monte Carlo Simulation (MCS) is used to apply the probabilistic approach. The operating characteristic of each component in the system is represented by the twostate model described by up and downstates, and the operating state of the whole power system can be obtained by considering the state of all components in the system and the uncertainty of weather conditions as shown in
Fig. 2
. The operating time in the upstate is called time to failure (TTF) and repair time in the down state is called time to repair (TTR). The TTF and the TTR can be expressed by the exponential distribution
[12]
.
Probabilistic TTC calculation.
State of all components using sequential MCS.
where
λ_{i} : failure rate of component i
μ_{i} : repair rate of component i
U : uniformly distributed random number
4. Numerical Analysis
 4.1 Deterministic Assessment of TTC
In order to show the effectiveness of the proposed algorithm, it has been tested on a 6bus 7line system, which is shown in
Fig. 4
.
IEEE 6bus 7line system.
In order to show the effectiveness of the proposed method the case is tested as follows:

Case A: Constraints in voltage magnitudes at buses and thermal limits at transmissionsnot including transient stability

Case B: Constraints in voltage magnitudes at buses and thermal limits at transmissionsincluding transient stability
The sequential quadratic programming (SQP) is used to determine the TTC of each case.
Table 1
and
Fig. 5
show the result of the TTC and the transient stability in each fault. In table the TTC level of base case, 133.45 MW, means the maximum power that can be transferred from source area to sink area in the base case that no outage happens to the system. When the transient stability is checked in Case A, the TTC level of line *34 fault is equal to 141.98 MW but transient stability is not satisfied in
Fig. 5 (b)
. In Case B, the TTC level of line *34 fault is reduced to 107.92 MW while transient stability is satisfied in
Fig. 5 (e)
. It is seen that the TTC level can be determined by the transient stability as well as bus voltage magnitude and line thermal limits. As a result, in case of line fault 45 the initial TTC level is equal to 66.59 MW. It is seen that the TTC of the test system not including transient stability is equal to 66.59 MW, the smallest value of TTC levels. The TTC of this test system is determined not by transient stability but by thermal limits in deterministic assessment of TTC.
TTC level with and without transient stability
TTC level with and without transient stability
Transient stability.
 4.2 Probabilistic assessment of TTC
In order to apply the probabilistic approach considering uncertainty of weather, the weather data are divided into normal and adverse weather for 1 year from Korea Meteorological Administration (KMA), which is in 2012 year.
Fig. 6
shows that the weather is divided into the normal and adverse weather states for about 8760 hours, where the number 0 and 1 represent the normal weather and adverse weather conditions, respectively. In adverse weather conditions, the failure rate of a component can be considerably larger than the normal weather condition. This paper assumes that failure rate in the adverse weather is ten times higher than that in the normal weather
[8]
.
Division of normal and adverse weather.
Using sequential MCS, The operating state of the system in the normal condition only and the condition that include adverse weather condition for 1 year is shown in
Fig. 7
and
Fig. 8
, respectively. The horizontal axis is the operating state of the system for 8760 hour and the vertical axis is the number of surviving components; for example, the number 5 means that two components are faulted among the six 7 components
[8]
. From
Fig. 8
, it is seen that the frequency of outage in the condition that include adverse weather is higher than for the normal weather condition. The sequential MCS can be taken to provide the probability distribution function (PDF) of TTC. Considering the normal weather condition,
Fig. 9
shows the system can withstand most transient faults in the range between 130 and 140 MW in probability.
System state data when 1 year is only normal weather.
System state data including adverse weather for 1 year.
PDF of TTC with the only normal weather.
Because the outages spread widely without special trend in
Fig. 7
, the more the TTC value in
Fig. 9
moved to large value side, the higher the probability of the outages is.
On the other hand, considering adverse weather in
Fig. 10
, surviving probability of system is lower than of normal weather. Because many outages occurred at the period affected by adverse weather in
Fig. 8
, the probability of the outages is high in lower portion of TTC value in
Fig. 10
.
PDF of TTC with including adverse weather.
If system is operating under the certain adverse weather condition, it is desirable to determine the TTC value in the range between 65 and 70 MW in probability in order to make the system stable. The reason is that the probability of TTC about 66 MW for the case with adverse weather is much higher than compared with the value in the normal weather for 1 year. Especially, it is seen that it is possible to operate power systems at lower TTC during the period such as a special weather statement, if the weather condition is considered in advance.
5. Conclusion
This paper presents a probabilistic method to evaluate the total transfer capability (TTC) by considering the energy margin and the uncertainty of weather conditions. In TTC determination the repeated power flow (RPF) method is used to maximize the incremental factor of load and generation and the transient energy margin method instead of the time simulation such as RungeKutta is used to check the transient stability. The weather condition that affects the system reliability is considered. As a result of considering weather effect and using probabilistic approach, the TTC for the adverse weather condition is lower than that of the normal weather condition. Especially, it is seen that it is possible to operate power systems at lower TTC during the period such as a special weather statement, if the weather condition is considered in advance.
Acknowledgements
This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF2011013D00041).
BIO
KyuHo Kim received his B.S., M.S. and Ph.D. degrees from Hanyang University, Korea, in 1988, 1990 and 1996, respectively. He is an Associate Professor in the Department of Electrical Engineering at Hankyong University, Korea. He was a Visiting Scholar at Baylor University for 20112012. His research interests include power system control and operation, optimal power flow and evolutionary computation.
JinWook Park received his B.S. degree in Electrical Engineering in 2001 from Semyung University, M.S. degrees from Hanyang University, Korea, in 2006. He is currently a Manger in the LG electronics, Korea. His research interests include Power system, distribution power system analysis and solar system.
SangBong Rhee received his B.S., M.S. and Ph.D. degrees from Hanyang University, Korea, in 1994, 1999 and 2004, respectively. He is currently an Assistant Professor in the Department of Electrical Engineering at Yeungnam University, Korea. His research interests include artificial intelligence applications, distribution power system analysis, operation, and control.
Sungwoo Bae received the B.S. degree from Hanyang University, Korea, and the M.S.E. and Ph.D. degrees from the University of Texas at Austin, USA, all in electrical engineering, in 2006, 2009, and 2011, respectively. From 2012 to 2013, he was a senior research engineer with Power Center at Samsung Advanced Institute of Technology. He is currently an Assistant Professor in the Department of Electrical Engineering, Yeungnam University.
KyungBin Song received his B.S. and M.S. degrees in Electrical Engineering from Yonsei University, Korea, in 1986 and 1988, respectively. He received his Ph.D. degree in Electrical Engineering from Texas A&M University, College Station, Texas in 1995. He is currently an Associate Professor in Electrical Engineering at Soongsil University, Seoul, Korea. His research interests include power system operation and control, power system economics, the optimization of the large scale systems, and the fuzzy system and its applications.
Junmin Cha received his B.Sc., M.Sc. and Ph.D. degrees from Korea University in 1989, 1991 and 1996 respectively. His research interest includes Fuzzy Applications, Probabilistic Production Cost Simulation, Reliability Evaluation, Outage Cost Assessment of Power Systems and Operation and Planning of Deregulated Power Market. Since 1996, he has been a faculty of Daejin University in Korea, where is now a professor.
Kwang Y. Lee received the B.S. degree in Electrical Engineering in 1964 from Seoul National University, M.S. degree in Electrical Engineering in 1968 from North Dakota State University, and Ph. D. degree in Systems Science in 1971 from Michigan State University. He was elected as a Fellow of IEEE in the January 2001 for his contributions to the development and implementation of intelligent system techniques for power plants and power systems control and as a Life Fellow of IEEE since January 2008. He has been working in the area of power plants and power systems control for over thirty years at Michigan State, Oregon State, University of Houston, the Pennsylvania State University, and the Baylor University, where he is Professor and Chairman of the Department of Electrical and Computer Engineering. His research interests include control, operation, and planning of power and energy systems; computational intelligence, intelligent control and their applications to power and energy systems, and modeling, simulation and control of microgrids with renewable and distributed energy sources.
1996
“Available Transfer Capability Definitions and Determination,” NERC Report
Rosehart W.
,
Canizares C.
,
Quintana V.
1999
“Optimal power flow incorporating voltage collapse constraints,”
IEEE PES Summer Meeting
1822, July 1999
2
820 
825
Ajjarapu V.
,
Christy C.
1992
“The Continuation Power Flow: A Tool for Steady State Voltage Stability Analysis,”
IEEE TPS
7
(1)
Yu X.
,
Singh C.
2004
“Probabilistic Analysis of Total Transfer Capability Considering Security Constraints,”
8th International Conference on PMAPS
Iowa State University
September 1216, 2004
Yu X.
,
Singh C.
,
Jakovljevic S.
,
Ristanovic D.
,
Huang G.
2003
“Total transfer capability considering FACTS and security constraints,”
Proc. 2003 IEEE PES Transmission and Distribution Conf. And Expo.
Sep. 2003
1
73 
78
Audomvongseree K.
,
Yokoyama A.
2004
“Consideration of an Appropriate TTC by probabilistic Approach,”
IEEE Transaction on Power System
19
(1)
Berizzi A.
,
Bovo C.
,
Delfanti M.
,
Merlo M.
,
Savino M.
2007
“A Monte Carlo Approach for TTC Evaluation,”
IEEE Transaction on Power System
22
(2)
Park J. W.
,
Kim K. H.
,
Lee H. S.
,
Kim J. O.
,
Shin D. J.
2005
“Assessment of Total Transfer Capability Considering Uncertainty of Weather,”
International Conference on Electrical Engineering
Kunming, China
Pai M. A.
1989
Energy Function Analysis for Power System Stability
Kluwer Academic Publishers
Fouad A. A.
,
Vittal V.
1992
Power System Transient Stability Analysis Using The Transient Energy Function Methods
Prentice Hall
138 
Kim S. N.
,
Kim K. H.
,
You S. K.
2002
“Realtime estimation of multiple series capacitor quantity for transient stability energy margin improvement,”
IEEE PES Winter Meeting
238 
243
Billinton R.
,
Allan R. N.
1996
Reliability Evaluation of Power System
Plenum Press
New York
Kim K. H.
,
Kim S. N.
,
Rhee S. B.
,
Lee S. K.
,
Song K. B.
2008
“Assessment of total transfer capability subject to transient stability energy margin,”
IEEE Transmission and Distribution Asia
238 
243