IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY2005 573 Two-Point Estimate Method for Quantifying Transfer Capability Uncertainty Chun-Lien Su, Member, IEEE, and Chan-Nan Lu, Senior Member, IEEE Abstract—A two-point estimate method is proposed in this paper to assess the power transfer capability uncertainty. This paper as- sumes that the uncertainty of the line parameters and bus injec- tions involved in transfer capability calculations can be estimated or measured and shows how to estimate the corresponding uncer- tainty in the transfer capability. Instead of using a large number of simulations as required in the Monte Carlo approach, for a system with uncertain parameters, the two-point estimate method uses calculations of transfer capability to quantify the uncertainty. The proposed method uses a numerical method to calculate the moments of the transfer capability. The moments are then used in the probability distribution fitting. Using the obtained transfer ca- pability uncertainty information and a desired level of reliability, an adequate transmission reliability margin can be determined for each transmission service. The proposed method can be used di- rectly with a deterministic computer program and it does not re- quire derivativesof the transfer capability. Test results of the pro- posed method are compared with those obtained from the Monte Carlo simulations and a truncated Taylor series expansion method. Index Terms—Point estimate method, transfer capability, trans- mission reliability margin, uncertainty. NOMENCLATURE Uncertain parameters. Transfer capability. Coefficient of variation. th moment of . Safety margin. Reliability margin. Uncertainty of transfer capability. Probability distribution of safety margin. Probability distribution of uncertain param- eter. Nonlinear transfer capability function. Weighting of the concentration. Reliability index. , , , Means of , , , and . , , , Standard deviation of , , , and . Coefficient of skewness of . Manuscript received January 21, 2004; revised July 14, 2004. This work was supported by the National Science Council of Taiwan under Grants NSC 92-2213-E-022-004 and NSC 92-2213-E-110-009. Paper no. TPWRS-00026-2004. C.-L. Su is with the Department of Marine Engineering, National Kaohsiung Marine University, Kaohsiung, 805 Taiwan, R.O.C. (e-mail: cls@mail.nkmu.edu.tw). C.-N. Lu is with the Department of Electrical Engineering, Na- tional Sun Yat-Sen University, Kaohsiung, 804 Taiwan, R.O.C. (e-mail: cnl@ee.nsysu.edu.tw). Digital Object Identifier 10.1109/TPWRS.2005.846233 I. INTRODUCTION A N ESSENTIAL contributing factor in a power system blackout incident is the heavy use of the transmission network in an operation environment for which it was never de- signed. The movement toward open-access of the transmission network has added considerable interest in quantifying electric power system transmission capabilities. To balance system re- liability and commercial demand, safety margins of the system are reserved to accommodate the uncertainties involved with the system operations. Transmission reliability margin (TRM) is defined by the North American Electric Reliability Council (NERC) as the amount of transmission capability necessary to ensure that the interconnected network is secure under a reasonable range of uncertainties in system conditions [1]. Available transfer capability (ATC) is determined as a function of increase in power transmission between different locations. ATC is total transfer capability (TTC) minus base case flow and appropriate margins. TTC can be obtained by linearized power flow [1], [2], continuation power flow [3], [4], repeated power flow [5], [6], security constrained optimal power flow and probabilistic analysis [7]–[9]. To take the uncertainties of power system operation condi- tions into account in computing ATC, several approaches have been proposed to assess the TRM. The first one is based on re- peated computations of TTC using variations in the base case data. This is a “Monte Carlo” statistical approach. A second is a single repeat computation of TTC using safety bounds reduced by a fixed percentage (e.g., 4%). A third is simply to reduce the base case TTC by a fixed percentage (e.g., 5%). A fourth is a probabilistic approach using statistical forecast error and other systematic reliability concepts [10], [11]. In [11], the cen- tral limit theorem is used and the transmission capability un- certainty is approximated by a normal distribution with a mean zero and a standard deviation. A point-to-point transfer capability can be expressed as a function of many parameters Transfer Capability (1) Uncertainty in the parameters causes uncertainty in the transfer capability. The uncertain parameters include factors such as generation dispatch, customer demand, network param- eters and topology [11]. In this study uncertainties of line pa- rameters and bus injections are considered. For a transmission service, we are concerned with ensuring that the amount of TRM reserved is sufficient to accommodate the maximum uncertainty associated with the transfer capability due to the uncertainty in the parameters. The problem of TRM 0885-8950/$20.00 © 2005 IEEE