Monitoring the First Frequency Derivative to Improve ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 2, MAY 2011

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Monitoring the First Frequency Derivative to Improve Adaptive Underfrequency Load-Shedding Schemes Urban Rudez and Rafael Mihalic, Member, IEEE

Abstract—The actual activation of underfrequency load shedding is rarely needed in a power system. However, this does not diminish its significance, as it is the last barrier before a system blackout if underfrequency conditions occur. Like in all other areas of application, a tendency also exists to maximize the efficiency of the underfrequency load-shedding scheme and to bring its operation to the limits. By default, the scheme should be able to stop a frequency fall under any circumstances. However, trying to lower the quantity of the disconnected load in the same process is the purpose of this paper. After the theoretical background is summarized, in terms of the reaction of individual generators in the islanded system to a certain power imbalance, a procedure is described showing how it is possible to adjust any of the adaptive load-shedding schemes to utilize as much primary frequency control as possible. As a reference, a theoretically optimal underfrequency load-shedding scheme is presented. Index Terms—Adaptive control, frequency response, power system islanding, power system protection, underfrequency load shedding.

I. INTRODUCTION HE frequency of a power system is determined by the rotating speed of the synchronous generators in the system. Every excursion of the rotating speed of an individual generator from its nominal value is caused by an imbalance between the and the electrical torque on mechanical torque the generator's shaft [1]. If is greater than , the rotor is smaller than , of the generator accelerates and if the rotor of the generator decelerates. Every imbalance between the generated and consumed electrical power in a system is reflected in the generator's torque imbalance. In the case that the generators start to decelerate, . However, if the primary frequency control increases the frequency decay is rapid, this might not be a sufficient measure to keep the system frequency within the allowable limits, as the speed of the primary frequency-control activation is limited. is required by disconnecting the power Therefore, lowering system loads. An automatic procedure that disconnects a certain quantity of loads is referred to as underfrequency load shedding (UFLS). Different kinds of UFLS protection exist: traditional, semi-adaptive, and adaptive [2]. The adaptive UFLS schemes operate in two steps:

T

Manuscript received February 09, 2010; revised April 09, 2010; accepted July 09, 2010. Date of publication August 12, 2010; date of current version April 22, 2011. This work was supported by a Slovenian Research Agency as a part of the research program Electric Power Systems, P2-356. Paper no. TPWRS-001012010. The authors are with the Faculty of Electrical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia (e-mail: [email protected]; rafael. [email protected]) Digital Object Identifier 10.1109/TPWRS.2010.2059715

in the system; — calculating the active power deficit into several load-shedding steps. — distributing the can be calculated on the basis of the measured The initial-frequency derivative of the power-system frequency, slightly after the moment when a power imbalance in the ) [3]. However, during such system system occurs (at disturbances, the generators are disturbed by inter-generator oscillations. Consequently, they do not decelerate at the same rate. This is the reason why the frequency of the center of calculation. Although an inertia (COI) is needed for the online COI frequency calculation is not a trivial task, it is a necessity as the locally measured system frequency does not provide enough information. Consequently, the COI frequency is already widely used in various adaptive UFLS schemes. can be calculated using a Using the COI frequency, the frequency-response model [4], which for is basically a swing equation [1]. This procedure has been often used by a number of authors [3], [5]–[7], [8]. By additionally considering the effect of the voltage dependency of the system loads, the can be determined much more accurately [9]. In such a case, a power-system blackout can be avoided. distribution among the different Many strategies of the number of shedding steps are possible [2], [6]. In this paper, five different distributions are considered. A procedure for an additional adjustment of the shedding steps is presented in order to minimize the disconnected load on account of utilizing as much primary frequency control as possible. This paper deals with the circumstances that arise when a certain part of the power system is instantly isolated from the rest of the network [10]. The term used for such circumstances is “the transition to island operation,” and for the isolated part of the power system “the power-system island” or just simply “the island.” This so-called “islanding” can be treated as an active and reactive power injection, which has the unavoidable consequence of a change in the load flow. Namely, in general, the islanded part of the network either imports or exports a certain amount of the active and reactive power before the islanding takes place. After a quick overview of the inter-generator oscillations is , presented given in Section II., the principle of calculating in [9], is tested in Section III. Different distributions of the among the shedding steps are discussed next. Then, the procedure for adjusting the shedding steps is presented. After the results of the testing, the conclusions are drawn. II. INTER-GENERATOR OSCILLATIONS For the reader's convenience, this section provides an overview of the theoretical background, which describes the generators' behavior during a transient caused by a sudden

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change in the system's power balance. An understanding of this theory is the basis for the construction of the procedure that is presented in Section V-B. The content of this section follows [1], where a more detailed and comprehensive explanation of the subject is presented. Fig. 1. Test system with two generators and one load bus.

A. Swing Equation of a Single Generating Unit The motion of a machine rotor can be mathematically formulated using the so-called “swing equation.” The swing equation for the th generating unit can be written as (1) where is the inertia constant of the th generator in secthe first time derivative of the electrical angular onds, the th generfrequency of the th generator in p.u., and ator accelerating torque in p.u. is defined as the kinetic energy that is stored in the rotating , masses of the generator at the mechanical rated speed . normalized on the apparent rated power of the generator The p.u. base for the frequency is the electrical rated speed of , which corresponds to the nominal system the generator is defined as the differfrequency. The accelerating torque and the electrical ence between the mechanical torque torque . Dividing by the p.u. base value for the torque (the and ) gives the . quotient of

Fig. 2. Nine-bus IEEE test system.

C. Test Systems Used to Support Theoretical Considerations Two test systems have been used in the presented study: first, a system with two generators connected to a single load bus, depicted in Fig. 1, and second, the nine-bus IEEE test system presented in [1] and depicted in Fig. 2.

B. Swing Equation of a Multimachine System

D. Power Imbalance Distribution at

A swing equation formulates an individual generating unit's rotor motion. Therefore, to derive a swing equation for a multimachine system (with generating units), it is reasonable to imagine an equivalent generating unit that describes the average behavior of all the generating units. This equivalent generating unit is called the COI. First, it is important to alter the individual inertia constants and use the common system base as follows:

Every change in the motion of the generator's speed is basically a change in its frequency. Therefore, the swing equation is a handy tool, not only for transient-stability studies, but also for islanding studies. In transient-stability studies, by making being numerically nearly equal to the the assumption of , a negligible error is made for the entire interval of the achieved rotor frequencies [1]. UFLS on the other hand should prevent the frequency from falling below 47.5 Hz [11]. Compared to the rated power-system frequency of 50 Hz, the maximum possible frequency deviation is therefore 2.5 Hz or 5%. This deviation is relatively small and does not have much influence on the UFLS scheme's effectiveness. However, it is important to be aware of its existence. At any given moment, the electrical power produced by the generators should be equal to the electrical power consumed by the network. This includes the electrical loads together with all the power-system losses. Every change in this balance that disturbs the steady-state operation of the power system is referred to as “a power imbalance.” If such a power imbalance causes the generators to decelerate, then a power deficit is in question. This paper deals with the consequences of power-deficit conditions in a power system. When an island is formed, the power imbalance disturbs the island generators. As the balance between the consumption and the power flowing from the generation buses is kept unchanged, the generators cover different portions of the power imbalance, with the total sum being equal to the total power imbalance. Just ), the power after the moment of the island formation (at imbalance is distributed among the generators with regards to their synchronizing power coefficients. Assuming the change in

(2) where is the common power-system base. In the paper, , a pre-fault sum of all the island loads is for selected for practical reasons. By applying the procedure that and is given in [1], the COI electrical angle frequency are given as follows: the inertia constant (3) (4) For frequencies that do not deviate much from its nominal value, an assumption is made that is numerically nearly equal . Consequently, the COI swing equation is to the (5) where based on

is the acceleration power of the COI in p.u., .

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the angle and the voltage happens only on the island bus , the for the generator bus is synchronizing power coefficient defined by (6) Equation (6) is derived for the classical model of the multimachine power system. In such a model, and represent the th the susand th bus voltages behind the transient reactance, ceptance between the buses and (including the generator's the conductance between the buses transient reactance), and , and the pre-fault angle difference between the buses and . E. Oscillations and Power Imbalance Distribution After the Transient In the continuation, a transient takes place, which can be referred to as an “inter-generator oscillation.” The duration of this oscillation depends on the damping level of the islanded power system. However, after the transient is finished, the power imbalance is distributed among the generators in a completely different manner—according to their : (7) is the total power imbalance in p.u. and where is the th generator's share of the power imbalance in p.u. The described manner in which the power imbalance is distributed among the generators can be confirmed by a simple dynamic simulation of the islanding of the test system in Fig. 1. For a graphical representation, the so-called locus diagram of a frequency derivative versus acceleration torque is used (Fig. 3). is the th generator's share of the torque Accordingly, imbalance in p.u. and is the total torque imbalance, which aggravates the COI. Before islanding, the power system finds itself in the steady state. In Fig. 3, this is marked by the operating point A. The accelerating torque of the COI and also the first frequency derivative of the COI are equal to 0. After some time, the islanding takes place (a fault). At the first instant after the fault (marked with the letter B in Fig. 3), the synchronizing power coefficients determine the total torque-imbalfor genance distribution among the generators [ for generator 2, the sum of both gives erator 1 and for the COI]. In the continuation, an the total inter-generator oscillation takes place. One particular moment during the oscillation is marked with the letter C. However, after the transient is finished (letter D), different inertia constants define the different steepnesses of the locus diagrams of the individual generators [see Fig. 3 and (1)]. The power-imbalance distribution among the generators in an island is determined by relation (7). However, the changing rate of the frequency gradient is the same for all generators—equal to the changing rate of the COI. According to [1], the inter-generator oscillation of a system natural oswith synchronous generators is composed of cillation frequencies. This can be clearly seen from Fig. 4, where the first frequency derivatives of all three generating units in the

Fig. 3. Power-imbalance distribution among the generators.

Fig. 4. First frequency-time derivative of the individual generators and the COI during the inter-generator oscillations.

nine-bus IEEE test system are depicted, along with the first frequency derivative of the COI frequency. However, renewable sources of energy, connected to the system through inverters, do not contribute to oscillations or system inertia. In transient-stability studies, the angles of the generator rotors are observed with respect to the COI. However, in this paper, the COI is not a reference value. On the contrary, the goal is to observe the changes in the COI frequency, compared to some other reference. For this reference, the frequency of a constantfrequency source (infinite bus) is chosen. At first glance, it can be assumed from Fig. 4 that the COI does not oscillate. However, this is not exactly the case. During the inter-generator oscillation, the power flow on the transmission network oscillates with the same frequencies as the generators. The power flow through a certain transmission element causes ohmic losses which in general differ from the ohmic losses of the other transmission elements. Consequently,

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Fig. 5. Oscillation of the COI during inter-generator oscillations.

a power-flow oscillation causes the ohmic-losses oscillation. It must not be forgotten that the power-system losses (including the internal generators' losses) are also a part of the power-system load. This can be seen from the lower graph in Fig. 5, which represents the changes in the loads together with the changes in the transmission-system losses. It is clear from the upper graph in Fig. 5 that the COI frequency does oscillate and that these oscillations are directly dictated by the oscillations in the transmission network's ohmic losses. It might happen that before the islanding, the power flow through certain lines in the island is rather small. However, after the island's formation, the load flow and the power-system losses might change substantially. III. ACTIVE POWER DEFICIT CALCULATION Considering the theory from Section II, in order to be able from the initial frequency derivative, the freto determine quency derivative of the COI must be used. Locally measured frequencies do not contain enough information. In [8], [12], and [13], special attention was drawn to concalculation. As sider the load's voltage dependence during a result, it has been shown in [9] that the frequency derivative alone might give misleading information regarding at the active power-deficit value due to the fact that the loads are, in general, voltage dependent. According to [9], a much more exact calculation of the deficit is possible by applying (8) where the functions

and

are defined as (9) (10)

and represent the th load's active power and the bus is the th load voltage in the steady state prior to islanding. bus's momentarily voltage, is the number of all the loads in the system, and is the factor depicting the active power dependence of the th load on the voltage variations. Let us now show the consequences of ignoring the load's . An exvoltage dependence (function ) while calculating treme case is selected for this purpose: the case where no primary frequency control is available in the island, depicted in Fig. 2. Next, a load shedding of the calculated value of the deficit is applied in a single step at the same instant when the calculation is made. Obviously, this is a theoretical method. However,

Fig. 6. Importance of considering the load's voltage dependence.

it will show the importance of using (8) for the deficit calculation. The frequency trajectory and the total load in the system are shown in Fig. 6 for both cases. Obviously, by shedding without considering , not enough loads are shed to stop the frequency falling. This can lead to a blackout. In contrast, by considering , more loads are shed and the frequency remains constant during the entire simulation time. IV. DISTRIBUTION OF THE SHEDDING AMOUNT is calculated, this value should be distributed After among the different shedding steps according to (11) where is the number of all the shedding steps and is , distributed to the th shedding step. Three the portion of main aspects have to be studied regarding this subject [14]: — defining the total number of shedding steps; — defining the amount of the individual shedding steps; — defining the conditions for each shedding-step activation. Many different suggestions can be found in the literature . In [5], the principle of shedding about how to distribute larger steps first is proposed in order to lower the frequency derivative as soon as possible. In this section, five different distributions are observed. For all five cases, the number of all the shedding steps is = 4 and the conditions for the activation of the individual steps are equal to the following threshold frequencies: 49.0 Hz, 48.8 Hz, 48.4 Hz, and 48.0 Hz. These threshold frequencies are currently used in the traditional UFLS scheme of the Slovenian power system. Table I summarizes all of the five observed distributions, where the values in the table . are in a percentage of The schemes are evaluated by comparing the total (final) load-shedding amount. The schemes are applied to the nine-bus IEEE test system by changing the pre-fault generation in such a way that different deficits are simulated. Of course, it is verified for each individual simulation that the frequency does not fall below 47.5 Hz. The results are shown in Fig. 7. The lower values of the active-power deficit reflect in the lower values of the initial frequency derivative. Consequently, when the deficit is low, the distributions with the smaller first few steps shed less load than the others (e.g., distributions 4 and

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TABLE I SUMMARY OF FIVE LOAD-SHEDDING DISTRIBUTIONS

Fig. 8. COI frequency trajectory by applying the theoretically optimal UFLS scheme.

Fig. 7. Total load-shedding amount of all five different load-shedding distributions for various power deficits.

5). Namely, the primary frequency control has enough time to contribute a significant share to lowering the deficit. However, at higher values of the deficit, these smaller initial steps have too small influence on the frequency derivative, which is in such a case in the range of higher values. A rapid frequency decline is not decreased until the higher steps are executed. This happens at lower frequencies (see Table I). The primary frequency control does not have enough time to contribute significantly and, consequently, the distributions with smaller initial steps shed more load than the others. A similar explanation can be applied to the schemes with higher values of the shedding steps in the beginning (e.g., distributions 1 and 2). When the deficit is low, too much load is shed. However, at higher values of the gradient, less load is shed compared to the distributions like 4 and 5. The effectiveness of distribution 3 is somewhere in between both groups (distributions 1, 2 and distributions 4, 5). It is important to notice the shape of the curves in Fig. 7. This shape can be described as a linearly increasing function of the deficit with additional discrete jumps. This is a consequence of is distributed into discrete shedding steps. In the fact that this way, the load is also shed discretely. In the following section, a procedure is explained about how to avoid these discrete jumps and to make the shedding amount a nearly linear function of the deficit. V. ADJUSTING THE SHEDDING-STEPS AMOUNT The active power deficit, as a consequence of an island formation, can be decreased by applying two possible measures: load shedding and the primary frequency control. Load shedding lowers the system load by disconnecting certain load buses, whereas primary frequency control gradually increases the generation of production units. It is reasonable to avoid disconnecting the load as much as possible. This can be done only by

applying as much of the primary frequency control as possible. However, when load shedding is required, the frequency drop is usually relatively rapid. On the other hand, the primary frequency control does not react quickly enough to cope with the frequency changes by itself. Therefore, in order to minimize the disconnected load, both measures should be strongly mutually dependent. As the speed of the primary frequency control depends on the generating unit type and is therefore more or less unchangeable, a procedure is shown in this paper about how to adapt the UFLS to the primary frequency control characteristics. A. Theoretically Optimal UFLS Scheme Let us assume that we are not able to influence the primary frequency-control response. Under such circumstances, the maximum quantity of primary frequency control can be utilized only if its reaction is completely known in advance. As this cannot be achieved in practice, this section discusses the “theoretically optimal UFLS scheme.” The more time the primary frequency control has for its utilization, the more of its available amount can be utilized. From this point of view, it is reasonable to shed a certain amount of load in a single step, exactly at the same time when the initial frequency derivative of the COI is measured. An example of such a shed has already been shown in Fig. 6. Next, the question arises regarding the amount of such a shedding step. To determine this amount, the lowest acceptable limit . In our case, for the system frequency should be known the value of Hz was chosen [15]. The theoretically optimal UFLS scheme should shed just enough load, so . the frequency trajectory of the COI just barely reaches A dynamic simulation of applying such a scheme is depicted in Fig. 8. Even though this is only a theoretical scheme, it can usefully serve as a reference for evaluating the UFLS scheme's effectiveness. Namely, the result obtained using this scheme is the target value of all the other adaptive UFLS schemes. B. Adjusting the UFLS Steps In this section, the procedure for adjusting the predefined load-shedding steps to the primary frequency-control reaction is given. Compared to [5], where the adoption of a fixed correction factor of 1.05 is suggested, the proposed procedure introduces a dynamic correction. The procedure is based on the following linearity concept. As already written, the instantaneous is a linear function of the instantaneous value of the

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Fig. 9. Monitoring the COI frequency derivative is the indicator of primary frequency-response utilization.

value of or the approximately linear function of the in. In addition, it has been shown in stantaneous value of Fig. 6 that in the case of no primary frequency control being from (8) lowers the instantaneous value utilized, shedding to a negligible level. Therefore, in general, the x % of is reflected in an x % lowering of . lowering of The requirement for applying such a procedure is monitoring throughout the underfrequency conditions. In this way, it is possible to retrieve information regarding the changes in between the two neighboring shedding steps. If a the certain change is noticed, the following shedding step can be adjusted accordingly. This is due to the fact that a change in between the steps is an indicator that a primary frethe . Consequency control has taken over a certain part of the . quently, the UFLS is not required to shed this portion of The situation is graphically shown in Fig. 9. The graph reprein sents the first frequency derivative of the COI the islanded nine-bus IEEE test system with 220 MW of active power deficit. Three shedding steps are applied, which have the . consequence of discrete changes in the The adjustment of each of the shedding steps is possible by between two neighboring monitoring the changes in shedding steps and comparing it to the initial (maximum) value . For example: the shedding step 1 (marked with a circled 1) can be altered (in this particular case lowered) acbetween the initially measured cording to changes in and in the first step. This change is marked by . Similarly, the shedding step 2 (marked with a becircled 2) can be altered according to changes in tween the first and the second shedding steps. This change is . Following the same pattern, is relevant for marked by adjusting the shedding step 3 (marked with a circled 3). Before the th shedding step is activated, the percentage is first calculated: change of (12)

%

Keeping in mind the presented linearity concept, the upcoming ( th) shedding step can be altered from its predefined value according to %

(13)

Again, following the already-mentioned linearity concept, an additional shedding step is introduced in order to increase the

Fig. 10. Explanation of the shedding-steps adjustment procedure.

reliability of the procedure. If the frequency due to some un, the following is calculated: foreseen events still reaches %

(14)

where is a frequency derivative measured at . At , a shedding is made in the amount of . of % It should be emphasized that can be either a positive or a can be either higher or negative value. Consequently, . A demonstrative example of applying the lower than described procedure is given in the following section. VI. TESTING THE PROCEDURE A shedding-step adjustment procedure was tested on a nine-bus IEEE test system, applying different predefined disinto individual shedding steps (Table I). A tributions of demonstrative example is given in Fig. 10, where the distribution 3 with equal shedding steps has been applied. The first graph depicts the COI frequency trajectory in Hz and the second graph, the total sum of the system load's active power in MW. In the third graph, the grey curve represents the predefined value of the shedding steps and the black curve, the adjusted value of the activated shedding steps, both in a percentage of the pre-fault system load. In the fourth graph, the voltage on node 1 of the test system is depicted with the black and the curve. The light-grey curve shows the . dark-grey curve the At the moment marked with the circled 0, an island is formed % of the pre-fault system load (400 MW). with a The frequency starts to decrease drastically with an initial

RUDEZ AND MIHALIC: MONITORING THE FIRST FREQUENCY DERIVATIVE

Hz/s. When the frequency reaches is measured and the first frequency threshold, the % is calculated. This means that has % risen by 5.22% since the island was formed. Applying (13), to the first shedding step is lowered from the initial 25% . However, this change is caused mostly by the 19.78% drop in the load bus voltages and the consequent change in the system load. It should be noted that in Fig. 10, the number of shedding steps is depicted as a percentage of the pre-fault . Therefore, equals 14% and system load equals 11.08% . After shedding the first step and when the frequency reaches is measured again. Using the second shedding step, is noticed between steps 1 and 2 (12), a drop in % . This is due to the fact that the load-bus % voltages started to rise and consequently the remaining load also started to rise. Obviously, this rise is greater than the contribution of the primary frequency control at this point. is observed, Nevertheless, even though a drop in (13) is still used to alter the second shedding step. The second to 25.17% shedding step is adjusted from an initial 25% . A similar situation happens at shedding step 3, as a is observed between steps 2 and 3 drop in % . A change in the third shedding step % to 26.83% . is required from an initial 25% A second major contribution to lowering the total load-shedding amount happens during the fourth shedding step. Between is observed % steps 3 and 4, a rise in % and a lowering of the upcoming shedding step is required from to 21.13% . As is clear from Fig. 10, an initial 25% the frequency derivative is almost completely annulated after the activation of this last step. A small frequency excursion below 48.0 Hz is due to not considering the power-system losses in . However, the frequency does not drop the calculation of much below this last frequency threshold. Of course, from the point of view of minimizing the total load-shedding amount, it . would be sensible to shift the last threshold more towards However, to keep the reliability of the scheme at a high level, it is advisable to resist this temptation. The consequence of such a philosophy is a wider gap between the theoretically optimal results and the results obtained with the proposed procedure in Fig. 13. The procedure for adjusting the shedding steps has been tested for the same five distributions that were presented in Section IV. The results are shown in Fig. 11, along with the results of applying the theoretically optimal UFLS scheme. Clearly, the curves are much more linear than the curves in Fig. 7. However, for distributions 3, 4, and 5, a sudden increase in the curve can be seen: in the case of the distribution 3 passing the deficit of 260 MW, in the case of the distribution 4 passing the 200 MW, and in the case of the distribution 5 passing the 140 MW. This is a confirmation that the guideline for larger shedding steps to be assigned first is sensible. It is not trivial to determine which of the distributions yields the best results. Namely, it depends on the demands of the system operator. Observing the complete tested interval of deficits, an average shed of the distribution 3 is lower than that

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Fig. 11. Total load-shedding amount for all five different load-shedding distributions, applying the procedure for adjusting the shedding steps.

Fig. 12. Total load-shedding amount of the distribution 3 compared to the theoretically optimal scheme results.

Fig. 13. Total load-shedding amount of the distribution 2 compared to the theoretically optimal scheme results.

of other distributions. However, at deficits greater than 280 MW, the distribution 3 sheds more load compared even to a case without adjusting the shedding step sizes (Fig. 12). The reason for this lies behind the fact that the first steps disconnected fewer loads than would be required to lower the gradient to an acceptable level. However, if a system operator excludes the possibility that higher deficits could happen, then distribution 3 would be selected as the most appropriate. If this is not the case, the authors suggest the distribution 2 as the best option. It sheds more loads at lower deficits compared to distribution 3, but provides good results also at higher deficits (see Fig. 13). According to these findings, the authors suggest towards asa slight deviation from the equally distributed signing larger steps prior to the smaller ones.

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Furthermore, it should be noted that less load is shed by using the proposed procedure. By applying the distribution 3, even more than 16% less of the total load is shed (without considering the deficits above 260 MW) and by applying the distribution 2, even more than 17% less. This corresponds to lowering the shedding amount by approximately 38% (distribution 3) and 29% (distribution 2). VII. CONCLUSION The adaptive UFLS schemes are based on measuring the initial system frequency derivative. This measurement is used to calculate the active power deficit in an island. However, during the transition to island operation the inter-generator oscillations appear in the islanded power system. The power imbalance that aggravates the generators is distributed among them in two different manners. At the moment of islanding, it is distributed according to their synchronizing power coefficient and later on according to their inertia constants. This is the reason why the locally measured system frequency does not give enough information about the system's total deficit value. Consequently, the frequency of the COI is to be used in adaptive load-shedding schemes. The calculated deficit value can be distributed among the different shedding steps in different ways. These shedding steps might vary in their number, shedding amount, and the conditions for their activation. Five different distributions were tested in this paper. The results showed that when applying any of these distributions, the total load-shedding amount is not a linear function of the deficit value. In order to lower the shedding amount, a procedure is given in the paper for adjusting the amount of the shedding steps. The purpose is to adapt the steps' amount to the primary frequency-control response. The procedure was applied to all five distributions. The result is an almost linear relationship between the load-shedding amount and the deficit. However, depending on the expected possible deficit range in the islanded system, different distributions yield different results. In islands where greater values of the deficit can be excluded, schemes with smaller steps prior to the larger ones (like distribution 4) or schemes with equal steps (like distribution 3) are sensible. However, if the possibility of a greater deficit exists, the authors suggest a slight deviation from an equal-step distribution towards assigning larger steps prior to smaller ones (like distribution 2). By applying the presented procedure, a quantity of disconnected load can be significantly reduced. Nevertheless, the predefined distributions of the deficit into individual shedding steps should be defined carefully. In any case, nowadays it should not be too difficult to test the predefined islands for possible ranges of power deficit. REFERENCES [1] P. M. Anderson and A. A. Fouad, Power System Control and Stability, 1st ed. Ames, IA: The Iowa State Univ. Press, 1977.

[2] B. Delfino, S. Massucco, A. Morini, P. Scalera, and F. Silvestro, “Implementation and comparison of different under frequency load-shedding schemes,” in Proc. IEEE Power Engineering Society Summer Meeting, 2001, Jul. 15–19, 2001, vol. 1, pp. 307–312. [3] V. V. Terzija, “Adaptive underfrequency load shedding based on the magnitude of the disturbance estimation,” IEEE Trans. Power Syst., vol. 21, no. 3, pp. 1260–1266, Aug. 2006. [4] P. M. Anderson and M. Mirheydar, “A low order system frequency response model,” IEEE Trans. Power Syst., vol. 5, no. 3, pp. 720–729, Aug. 1990. [5] P. M. Anderson and M. Mirheydar, “An adaptive method for setting underfrequency load shedding relays,” IEEE Trans. Power Syst., vol. 7, no. 2, pp. 647–655, May 1992. [6] S. J. Huang and C. C. Huang, “An adaptive load shedding method with time-based design for isolated power systems,” Int. J. Elect. Power Energy Syst., vol. 22, no. 1, pp. 51–58, Jan. 2000. [7] H. You, V. Vittal, and Z. Yang, “Self-healing in power systems: An approach using islanding and rate of frequency decline-based load shedding,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 174–181, Feb. 2003. [8] M. S. Pasand and H. Seyedi, “New centralized adaptive under frequency load shedding algorithms,” in Proc. Conf. Power Engineering, 2007 Large Engineering Systems, Oct. 10–12, 2007, pp. 44–48. [9] U. Rudez and R. Mihalic, “Analysis of underfrequency load shedding using a frequency gradient,” IEEE, Trans. Power Del., to be published. [10] U. Rudez, V. Azbe, and R. Mihalic, “The application of a frequency gradient for underfrequency load shedding,” Electrotech. Rev., vol. 75, no. 3, pp. 155–161, 2008. [11] IEEE Power Engineering Society: IEEE Guide for the Application of Protective Relays Used for Abnormal Frequency Load Shedding and Restoration, IEEE Std. C37.117TM-2007, Aug. 24, 2007. [12] M. Etezadi-Amoli, “On underfrequency load shedding schemes,” in Proc. 22nd Annu. North American Power Symp., 1990, Oct. 15–16, 1990, pp. 172–180. [13] A. Li and Z. Cai, “A method for frequency dynamics analysis and load shedding assessment based on the trajectory of power system simulation,” in Proc. 3rd Int. Conf. Electric Utility Deregulation and Restructuring and Power Technologies, 2008 (DRPT 2008), Apr. 6–9, 2008, pp. 1335–1339. [14] C. Concordia, L. H. Fink, and G. Poullikkas, “Load shedding on an isolated system,” IEEE Trans. Power Syst., vol. 10, no. 3, pp. 1467–1472, Aug. 1995. [15] UCTE, Operation Handbook, “Appendix 1, Load-frequency Control and Performance,” Dec. 14, 2009. [Online]. Available: http://www.entsoe.eu/fileadmin/user_upload_library/publications/ce/oh/appendix1_v19.pdf. Urban Rudez received the B.Sc. degree from the University of Ljubljana, Ljubljana, Slovenia, in 2005. After finishing his graduate study, he worked with the Korona company in Ljubljana for two years as a System Engineer in the Department for Power Engineering. In 2007, he joined the Department of Power Systems and Devices at the Faculty of Electrical Engineering, where he still works as a Junior Researcher. His area of interest includes power system analyses and FACTS devices.

Rafael Mihalic (M’95) received the Dipl.Eng., M.Sc., and Dr.Sc. degrees from the University of Ljubljana, Ljubljana, Slovenia, in 1986, 1989, and 1993, respectively. He became a Teaching Assistant in the Department of Power Systems and Devices, Faculty for Electrical and Computer Engineering, University of Ljubljana, in 1986. Between 1988 and 1991, he was a member of the Siemens Power Transmission and Distribution Group, Erlangen, Germany. Since 2005, he has been a Professor at the University of Ljubljana. His areas of interest include system analysis and FACTS devices. Prof. Mihalic is a member of Cigre (Paris, France).