# Analysis of Load Frequency Control Performance Assessment Criteria

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3 522 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 3, AUGUST 2001, the values of the smoothed variable within a desired range. For the probabilistic structure, a natural metric is the mean square value to measure how closely the values of are bounded within the desired range. We, thus, propose as the interconnection LFC performance measure. We specify its bound to be and formulate the interconnection LFC performance criterion as Such a metric, however, is not practical for several reasons. One is that criterion does not consider the contribution of the various control areas to the interconnection performance. Another is that the criterion fails to use much of the information in the monitored measurements. In actual interconnected power system operations, each control area shares in the responsibility for LFC. Since some areas may have undesirable control performance, we need to construct for all the areas specific LFC performance metrics and criteria to ensure that each area is performing its appropriate share in LFC. B. Area Control Performance Criteria As the first step in monitoring each area s performance, we need to know how each area responds to the interconnection frequency error based on available measurements. To assess how the behavior of two r.v. s is related, we use their correlation as an indicator. A positive correlation indicates that the two r.v. s move in similar direction, while a negative correlation indicates the opposite [10]. We propose to monitor the correlation of and. The frequency error is a function of the interconnection s actual-scheduled generation imbalance [2] and is the actual-scheduled generation imbalance of area. Consider the case when has a positive value. In such a case, there is overgeneration throughout the interconnection, i.e., total generation exceeds the total load in the interconnection. If has a positive value, this area has more actual than scheduled generation and, consequently, it does not help the interconnection to reduce the interconnection overgeneration and restore the frequency to the synchronous value. Then, has a positive correlation with, i.e.,, and area s control performance does not help to restore the frequency to its scheduled value. On the other hand, if has a negative value, this area has lower actual than scheduled generation. Then, has a negative correlation, i.e.,, and this situation helps the interconnection to reduce the overgeneration and restore the frequency to the synchronous value. Then, positive [negative] indicates the area s undesirable [desirable] control performance. A symmetric situation occurs for the case of. We propose to use the correlation for each as a metric of each area s control performance. To ensure that the positive correlation is not excessive, we need to impose an upper bound on, whereas we need not care about the negative correlation since it indicates a desirable performance. A further consideration is that too wide of an excursion of the ACE from zero is not desirable in every single area. We (2) again use a probabilistic measure to assess how well each control area can contain the value of ACE within a specified range. Given a specified tolerance for area, let be the event that has an absolute value less than or equal to. Then, is the confidence level that the event happens [9]. The closer the confidence level is to 1, the more likely is the event to occur. We propose as an additional metric of each area s control performance. We want this value to exceed a specified confidence level. Using the proposed metrics, we construct for each area s control performance criteria and, where, (3) (4) We show that the criterion is a sufficient condition for the criterion. In addition, we prove that under assumptions that generally hold for windows whose length minutes, the criterion is equivalent to the criterion. The proofs are given in the Appendix. We used the development in [6] to motivate the formulation of the criteria in equations (3) and (4). Consider the discrete time random process (d.r.p.) obtained for the time set. This is a set of i.i.d. r.v. s. For large, the d.r.p. is ergodic [9]. The mean, the variance and the mean square value of may be estimated from any sample path of the process [9]. For example, the mean of may be estimated by, where is the observed value of from any sample path [9]. This characterization of consequently applies to the r.v. s,,. We make use of this important characteristic of ergodic processes in the application of the proposed criteria. The framework we constructed in this section is based on the application of basic notions of r.v. s and random processes (r.p. s) to LFC. The judicious use of the measurement set and smoothing allows the specification of implementable criteria to assess the performance of each area in discharging its LFC responsibility. The proposed criteria make use of natural probability measures such as correlation and confidence level of r.v. s and take advantage of the ergodic d.r.p. characteristics. We use this framework to analyze the control performance assessment criteria in the section below. III. RELATIONSHIP OF THE PROPOSED AND NERC CRITERIA The proposed criteria and are formulated to be rather general criteria for control performance assessment. In this section we show that, in particular, we can derive from them the NERC CPS1 and CPS2 criteria [5] as special cases. In addition, we derive an important new result showing that under certain conditions that generally hold, the satisfaction of CPS1 implies that CPS2 is also satisfied. As such, CPS2 is a redundant criterion. Once CPS1 is satisfied, it becomes unnecessary to check CPS2. The starting point of our analysis is the framework that was constructed in Section II. Note that the elements of the d.r.p. s and

4 GROSS AND LEE: ANALYSIS OF LOAD FREQUENCY CONTROL PERFORMANCE ASSESSMENT CRITERIA 523 are the i.i.d. r.v. s and, respectively. Moreover, for, and are statistically independent. Consider the r.v. s,. It follows that is a set of i.i.d. r.v. s and consequently is an ergodic d.r.p. For each, may be estimated from any sample path of the d.r.p. [9]. Let be an arbitrary sample path. For sufficiently large, may be estimated for each by [9]. Then, criterion is satisfied if and only if Let the window length be 1 minute and the interval length be 1 year, then the number of 1-minute windows within the one-year interval is sufficiently large. Let, the specified tolerance in the NERC CPS1 [5]. The expression in equation (5), then, is the same as the NERC CPS1 criterion. We adopt the notation for, where is expressed in minutes, is given in years and is the tolerance in mhz. We use this notation also for. Then, CPS1 may be expressed as. Next, we consider the criterion in equation (4). We define the r.v. if otherwise for a given value of. Then, the mean of is and the criterion in equation (4) may be restated as Since the r.v. s, are i.i.d., the r.v. s, are also i.i.d. and the d.r.p. is therefore ergodic. Let and be values of the r.v. s and, respectively. Then, if otherwise The mean is estimated by. We may interpret this estimate to be the proportion of the windows covering the interval of length which have the absolute value of ACE smaller than or equal to. Then, criterion of equation (4) is satisfied if and only if Let and, the specified tolerance value in the NERC CPS2 [5]. Then, in equation (9) is the same as CPS2. We next consider the relationship between and. The mean of for each is independent of the window length since (5) (6) (7) (8) (9) (10) Furthermore, for the variance, we have (11) Here, is the cardinality of the window. We use equations (10) and (11) to determine Since, the grid is uniform so that and it follows that (12) (13) Consider the criterion for two distinct windows and tolerances : (14) (15) Since the criterion is a sufficient condition for the criterion and the CPS1 is equivalent to the criterion, it follows that the CPS1 is a sufficient condition for the criterion. Using equation (13) and (15), holds if and only if We introduce the following condition where, (16) (17) (18) It follows that under this condition,. If CPS1 is satisfied,, where, the second inequality is due to the condition in equation (17). Consequently, the relationship of equation (16) holds so that is also satisfied. Thus, under the condition in equation (17), if each area satisfies CPS1, then and are also satisfied. The CPS2 criterion provides no additional assurance or insight in terms of control performance. Then, under the condition in equation (17) there is no need to monitor two separate criteria CPS1 and CPS2 and the monitoring of CPS1 suffices. IV. NUMERICAL STUDIES We carried out a number of numerical studies to ascertain whether the condition in equation (17) for redundancy of CPS2 can be determined. We performed the studies on a number of NERC interconnections. We provide a representative sample here for the following four interconnections: Eastern Interconnection, WSCC, ERCOT and Hydro-Quebec. Using the values of the NERC-specified and for each interconnection [5], we calculate the corresponding values. We estimate each interconnection s expected value of frequency error

5 524 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 3, AUGUST 2001 TABLE I DATA FOR THE NUMERICAL STUDIES The framework brings considerable new analytic insights into the LFC performance assessment problem. It can be used for the formulation of new control performance criteria and for the analysis of the measurement and monitoring requirements for LFC. This is particularly important as LFC is becoming an unbundled ancillary service in the new open access transmission regimes. APPENDIX using the frequency data obtained from sufficiently large number of time instants in each interconnection. The selected periods for the inspection are one year from May 1998 to April 1999 in the Eastern Interconnection [5], the year 1998 in WSCC, the month of January 1998 in ERCOT and the month of August 1997 in Hydro-Quebec [11]. We developed the results in the previous section under the assumption of year. However, given that the frequency errors are sampled every 1 3 seconds in these interconnections, one month is sufficiently long to provide a statistically meaningful data set for the evaluation of the estimate. Due to the ergodic nature of the r.p.s in the control performance criteria, the single month data can be tested effectively to ascertain the redundancy of the CPS2 criterion. Table I gives the values of and specified by the NERC [5], the calculated values of and the estimated values of for each interconnection. We can see that the condition in equation (17) is satisfied by each of the four interconnections for the selected periods. In these cases, CPS2 is a redundant criterion and the control areas do not need to monitor whether CPS2 is satisfied. As such, the areas accomplish their objective for the LFC once CPS1 is satisfied. We prove the fact that the satisfaction of the area control performance criterion implies the satisfaction of the criterion. In addition, we show the criterion to be equivalent to under assumptions that generally hold for window length. We start with the definition of the average ACE variables for each window: (A.1) Let be the number of control areas in the interconnection. Since the sum of the net interchange errors over the interconnection is zero, (A.2) with. Suppose each area satisfies the criterion of equation (3), i.e.,,. Then, for all the areas V. CONCLUSION In this paper, we constructed a random process based LFC performance assessment framework. This framework is effective in dealing with the random characteristics of the variables of interest in LFC. The framework provides a solid analytic basis for the formulation, analysis and evaluation of LFC performance criteria. We used the smoothed variables over non!overlapping windows to get more insight on the control performance assessment issues. The use of statistically meaningful metrics for the area control performance assessment led to the formulation of rather general performance criteria and. The ergodic nature of the random processes permits the evaluation of the proposed metrics using chronological data. The proposed criteria are general and include the criteria CPS1 and CPS2 as special cases. Moreover, the application of the framework shows that the satisfaction of CPS1 implies that CPS2 is also satisfied under a certain condition. Consequently, CPS2 criterion provides no additional insight in terms of control performance and the monitoring of CPS1 suffices. (A.3) It follows that and so the criterion is satisfied. Consequently, the criterion is a sufficient condition for the criterion. Next, we show that the criterion is equivalent to the criterion under a set of assumptions. We introduce the following two assumptions for window length : i) The r.v. s and of two different control areas are independent; and, ii),. These assumptions are reasonable for the interconnections of interest. In fact, analysis of actual data shows that for the control areas in North America, for windows, and of two different control areas are independent r.v. s [6, pp. 6 7]. It is also shown in [6] that,, is sufficiently small for window length

6 GROSS AND LEE: ANALYSIS OF LOAD FREQUENCY CONTROL PERFORMANCE ASSESSMENT CRITERIA 525 so as to be negligible. Consequently, for such window length the assumption (A.4) introduces no loss of generality. Let. Then, It follows from assumption ii) that that the relationship (A.9) holds if and only if Since the assumptions i) and ii) imply that for that so (A.12), it follows using equation (A.2) to obtain the last equality. Thus, the relationship (A.12) holds if and only if (A.5) Here we used equation (A.4) and the i.i.d. property of. We also determine the mean of as (A.6) Now, as becomes sufficiently large, in equation (A.5) tends to zero. Since var and are positive and var, the fact that tends to zero implies that necessarily both var and tend to zero. It follows from equation (A.6) that also tends to zero. Therefore, the introduction of assumption ii) does not lead to any loss of generality. We first rewrite the criterion as where, we substitute equation (4). Let (A.7) in (A.8) Then,, are i.i.d. r.v. s with standard normal distribution, i.e., [8]. We rewrite relationship (A.7) in terms of as (A.13) which is the same as the criterion. Consequently, the criterion is equivalent to the criterion under the two assumptions introduced above. Since the criterion is a sufficient condition for the criterion as shown above, the criterion is also a sufficient condition for the criterion under these two assumptions. REFERENCES [1] NERC IOS Working Group, Defining Interconnected Operations Services Under Open Access: Final Report, NERC, Mar [2] P. Kundur, Power System Stability and Control. New York, NY: Mc- Graw-Hill, 1994, pp [3] D. N. Ewart, Automatic generation control: Performance under normal conditions, in Proceedings of System Engineering for Power: Status and Prospects, Henniker, NH, Aug. 1975, pp [4] N. Jaleeli, L. VanSlyck, D. Ewart, L. Fink, and A. Hoffmann, Understanding automatic generation control, IEEE Trans. Power Systems, vol. 7, no. 3, pp , Aug [5] NERC. Operating Manual. [Online]. Available: [6] N. Jaleeli and L. VanSlyck, Control performance standards and procedures for interconnected operation,, EPRI TR , Apr [7], NERC s new control performance standards, IEEE Trans. Power Systems, vol. 14, no. 3, Aug [8], Tie-line bias prioritized energy control, IEEE Trans. Power Systems, vol. 10, no. 1, pp , Feb [9] H. Stark and J. Woods, Probability, Random Processes and Estimation Theory for Engineers. Englewood Cliffs, NJ: Prentice Hall, [10] R. D. Yates and D. J. Goodman, Probability and Stochastic Processes. New York, NY: John Wiley & Sons, 1999, pp [11] NERC, private communication providing the data for the WSCC, ERCOT and Hydro-Quebec systems. (A.9) Since for the standard normal distribution [8], the relationship (A.9) holds if and only if (A.10) George Gross (F 88) is Professor of Electrical and Computer Engineering and Professor, Institute of Government and Public Affairs, at the University of Illinois at Urbana-Champaign. His current research and teaching activities are in the areas of power system analysis, planning, economics and operations and utility regulatory policy and industry restructuring. His undergraduate work was completed at McGill University, and he earned his graduate degrees from the University of California, Berkeley. He was previously employed by Pacific Gas and Electric Company in various technical, policy and management positions. or, equivalently if and only if (A.11) Jeong Woo Lee received the B.S. and M.S. degrees in electrical engineering from Seoul National University in 1994 and 1996, respectively. He is currently working toward the Ph.D. degree in University of Illinois at Urbana-Champaign.

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