1324 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL....

1324 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 2, MAY 2013

Sensitivity Analysis of Load-Damping Characteristicin Power System Frequency Regulation

Hao Huang, Student Member, IEEE, and Fangxing Li, Senior Member, IEEE

Abstract—The smart grid initiative leads to growing inter-ests in demand responses and the load models, especially thefrequency-sensitive loads such as motors. The reason is thathigh-penetration controllable load may have substantial impacton system frequency response (SFR). However, the effect of thefrequency-related load-damping coefficient is still not completelyunderstood. This paper investigates the effect of frequency-sensi-tive load on system frequency using typical SFR model. Theoreticanalyses based on transfer functions show that the frequencydeviation under a different load-damping coefficient is relativelysmall and bounded when the power system is essentially stable;while the frequency deviation can be accelerated when a powersystem is unstable after disturbance. For the stable case, thelargest frequency dip under a perturbation and the correspondingcritical time can be derived by inverse Laplace transformationusing a full model considering load-damping coefficient. Further,the error in evaluating the load-frequency coefficient gives thelargest impact to frequency deviation right at the time when thelargest frequency dip occurs. Multiple-machine cases and auto-matic generation control (AGC) are also included in the analyseswith verifications by simulation studies. The conclusion can beuseful for system operators for decision-making of load control orinterruption.

Index Terms—Automatic generation control (AGC), demand re-sponse, frequency drop, frequency sensitive load, load control, sen-sitivity analysis, system frequency response (SFR).

NOMENCLATURE

Angular frequency deviation.

Laplace transformation of .

Frequency deviation in Hz.

Load change perturbation, usually considered asa step function.Governor time constant.

Steam chest time constant.

Reheat turbine time constant.

High pressure power fraction of reheat turbine.

Generator inertia constant.

Manuscript received February 04, 2012; revised February 23, 2012 and June15, 2012; accepted July 15, 2012. Date of publication October 03, 2012; dateof current version April 18, 2013. This work was supported by the NSF undergrant ECCS 1001999. Also, this work made use of Engineering Research Center(ERC) Shared Facilities supported by the NSF/DOE ERC Program under NSFAward EEC-1041877 and the CURENT Industry Partnership Program. Paperno. TPWRS-00102-2012.The authors are with the Department of Electrical Engineering and Com-

puter Science, The University of Tennessee, Knoxville, TN 37996 USA (e-mail:[email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRS.2012.2209901

Load-damping coefficient.

Governor speed regulation.

Number of generations.

Integral controller gain.

Unit step function, is the start point ofdisturbance.Small value less than 1.

Large value greater than 1.

Phase angle of .

Unit-less (normalized) frequency sensitivityfunction of .

I. INTRODUCTION

T HE frequency of a power system is a very important per-formance signal to the system operator for stability and

security considerations. The desired power system frequencyshould stay within a very small, acceptable interval around itsnominal value. Otherwise, the operator needs to take relevantactions immediately. In the past decades, there are many re-search works on power system frequency regulation [1]–[4].Since the system frequency is essentially related to real powerbalance, it is natural to control real power output in the genera-tion side such as using the automatic generation control (AGC)system. This is indeed one of the very successful control appli-cations in the power system operation in the past decades.In recent years, the increasing stress in the transmission

system may limit the effective power transfer from generationto load. Also, the penetration of intermittent renewable re-sources continues growing. Thus, the frequency and the relatedstability issue are being re-examined under this new paradigm[5]. Non-conventional means, such as direct load control anddynamic pricing under the smart grid initiative, have attractedmany research interests as an alternative solution for balancingservice and frequency regulation [6]–[8], especially under ahigh penetration of load control.Some early works on the load shedding or load control topic

is to set up the well-known generation swing equation first, andthen to employ classic tools in control theory, such as transferfunction [1]–[4] and state space method [3], into the swingequation to find out the amount of load to shed. In [1] and [2], aclassic model, system frequency response (SFR), is introduced,and it is still accepted broadly. Also, in [2] and [9], differentimplementations of adaptive under frequency load shedding(UFLS) are presented. In [10], a load shedding optimizationscheme is presented. In [11], a classic closed-loop PID controlstrategy is implemented to regulate power system frequency.

0885-8950/$31.00 © 2012 IEEE

HUANG AND LI: SENSITIVITY ANALYSIS OF LOAD-DAMPING CHARACTERISTIC IN POWER SYSTEM FREQUENCY REGULATION 1325

In [12], a general-order SFR model with load shedding schemeis proposed to produce a closed-form expression of frequencyresponse. In [13], a method to determine the frequency stabilityborder for UFLS is presented. In [14], an SFR analysis approachsuitable for normal and contingency operation conditions isproposed.While many previous works were done to find the effect on

the system frequency due to external disturbances, such as alarge generator drop or a large load connection [1]–[4], [9], thereis little work on the effect of the intrinsic load characteristics.With the increasingly large amount of frequency-sensitive loadfor frequency regulation [6]–[8], [15] and the increasing inter-ests in load models [16], it is necessary to investigate the impactof load characteristics, namely, the load-damping coefficient, ,on the system frequency regulation. However, the load-dampingcoefficient is still not fully understood and usually assumed as aconstant from operational experience. Also, it may be highlyvariable under different operating points. In particular, undersmart grid initiative with high-penetration controllable loads,the interrupted loads should consist of lots of motor loads whichhave a significantly different load-damping coefficient than therest of the loads. Therefore, this paper is aimed to study the im-pact of variation of such coefficient, namely, ,on SFR. Byintuition, one may always perform several dynamic case studieswith various coefficient values; however, it is always desirableto have a fast and efficient model without repeated case studies.With the above motivation, this paper presents an efficient

analytical method to study the impact of the load-dampingcoefficient as well as a mathematical approach to derive thelowest frequency dip. Therefore, the system operators mayhave a fast assessment of different scenarios of load-frequencycharacteristics to understand the potential frequency deviationunder various emergencies, if the load-damping coefficientis different from the estimated value. The study in this paperindicates that the characteristic of frequency-sensitive load hasan important effect on SFR when the system is unstable, whichmeans the frequency protection devices may trip quicker thananticipated. Meanwhile, the study also shows that the externaldisturbance may dominate the load frequency characteristicswhen the system is stable; hence, the load frequency charac-teristic has much less impact on stability when the system isessentially stable.The remaining part of this paper is organized as follows: the

transfer function derivation, the multiple-machine case, and theeffect analysis of frequency-sensitive load-damping coefficientare shown in Section II; frequency stability analysis using thetotal differential equation, including the largest frequency dipderivation and the proof of alignment of the maximum sensi-tivity and the largest frequency dip, is presented in Section III;the simulation results are given in Section IV; and the con-cluding remarks are given in Section V.

II. SENSITIVITY FUNCTION DERIVATION

A. Single Machine (SISO) System—Without AGC

Here we consider a complete block diagram of the load-fre-quency control (LFC) for a simple single machine system, or

Fig. 1. Load frequency control diagram with input and output.

single input-single output (SISO) system, in Fig. 1[4]. Here themachine model considers a reheat turbine, which is typical forfrequency control, for illustrative purpose. Similar analysis andconclusions can be extended to other turbines like hydraulicones. With commonly adopted hypothetic assumptions, this isthe most simplified model. If a system is a multiple-machineand multiple-load system, it can be converted to an approxi-mate SISO system using generation and load aggregation [1],[17], [18]. If there is no AGC in this system, the closed-looptransfer function relating the fixed load step change, ,which is commonly assumed for LFC, to the angular frequencydeviation from nominal reference frequency 60 Hz, , isshown as follows:

(1)

By (1), the stability of this frequency-regulation system canbe tested by Routh-Hurwitz array or root locus. And the outputof angular frequency deviation can be obtained as

(2)

The proposed sensitivity analysis of the load-damping coeffi-cient, , is to calculate . As previously stated, thisshows the potential frequency variation when the actual valuediffers from the estimated value, or is assumed to change con-tinuously in a time period due to load control programs. Thisis important since the value is usually obtained empirically.Thus, the growing penetration of demand response draws theresearch interest on the impact of the value.Taking partial derivative of in (2), we can obtain the sen-

sitivity of the frequency deviation, , w.r.t. the load-dampingcoefficient, , as follows:

(3)

Then , the unit-less sensitivity function of , is derivedby its definition as follows:

(4)

B. Single Machine (SISO) System—With AGC

Consider the complete LFC block diagram of a simple singlemachine system with AGC in Fig. 2 [3]–[4]. The closed-loop


Fig. 2. Equivalent block diagram of AGC for an isolated power system.

transfer function relating the load change to the angularfrequency deviation is as follows:

(5)

Similar to (1), the stability of this frequency-regulationsystem with AGC shown in (5) can be tested by Routh-Hurwitzarray or root locus. The output of angular frequency deviationcould be obtained as

(6)If we take the partial derivative of in (6), the sensitivity ofw.r.t. can be written as

(7)

Then , the unit-less sensitivity function of , is derivedby its definition as follows:

(8)

C. Multiple-Machine System—No AGC

For the case of multiple generation machines without AGCas shown in Figs. 3 and 4, the transfer function is given by

(9)

The output of angular frequency deviation could be obtainedas

(10)

Fig. 3. LFC block diagram of multiple generation machines case with inputand output .

Fig. 4. LFC equivalent block diagram of multiple generation machines case.

Thus, the sensitivity of w.r.t. is given by

(11)

Then , the unit-less multiple-machine sensitivity functionof without AGC considered can be written as

(12)

D. Multiple-Machine System—With AGC

For the case of multiple generation machines with AGC asshown in Figs. 5 and 6, the transfer function is given by

(13)


Fig. 5. Block diagram of AGC for an isolated power system with multiple gen-eration machines case.

Fig. 6. Equivalent block diagram of AGC for an isolated power system withmultiple generation machines case.

The output of angular frequency deviation could be obtainedas

(14)

Thus, the sensitivity of w.r.t. is given by

(15)

Thus, , the unit-less multiple-machine sensitivity functionof with AGC considered can be written as

(16)

Note that (3), (7), (11), and (15) have the same formulation ofthe sensitivity function. Similarly, (4), (8), (12), and (16) showthe same formulation of the unit-less sensitivity function.

Fig. 7. SISO LFC block diagram with input and output andvaried load-damping coefficient .

III. STABILITY ANALYSIS USING TOTALDIFFERENTIAL EQUATION

A. Total Differential Equation for Frequency Deviation

The angular frequency deviation is only consideredto be related with external disturbance for simplicity inearly researches [1]–[4]. Its differential equation is as follows:

(17)

However, in (17), the effect of the load-damping coefficientin this SFR model is ignored. This may not give complete

information because the interrupted load may have a differentload-damping coefficient than the rest of the loads. This ishighly possible because many times interrupted or shed loadsare induction motor loads which have a value different fromother types of loads. Also, may be evaluated based on anout-of-date profile of load characteristic. In other words, thefrequency variation should be a function of andrather than only as shown in Fig. 7. Thus, it is interestingto investigate the impact of the load-damping coefficient.With the assumption that and are mutually inde-

pendent, (17) should be modified to include as follows:

(18)

Combining either (3), (7), (11), or (15) with (18), we have

(19)Furthermore, taking partial derivative for of equa-

tion either (2), (6), (10), or (14), we have

(20)

Combining (20) with (19), we have

(21)In order to have its time domain description, Laplace inverse

transformation is applied to (21). Thus, we have

(22)


Integration of (22) gives

(23)

Another interesting point should be mentioned from (23) isabout the effect of the load-damping coefficient on the sta-bility analysis in this SFR model in the next subsection.

B. Stability Analysis

1) When the Power System is Essentially Stable: If a powersystem is stable after disturbance, then its Laplace characteristicfunction’s poles are all located on the left half plane in s-do-main. That means the finite time-domain input wouldnot produce infinite time-domain output . It sufficesto say that the norm of the transfer function is bounded, i.e.,

for , from the perspectiveof control theory. Furthermore, from the perspective of powersystem design, should be in a small finite range sincethe system is essentially stable. The bound of the frequencydeviation is analyzed next.From (23), if we consider a step function, by triangle

inequality, we have

(24)

where . Note, when.

Apparently, the bound of from the traditional modelignoring the impact of is given by

(25)

The difference between the new model and the conventionalmodel is . This means the frequency deviationunder both models are bounded, though by different boundaries.2) When the Power System is Essentially Unstable: If the

power system is unstable, then some of its Laplace character-istic function’s poles are located on the right half plane in thedomain. That means the finite time domain input pro-duces infinite time domain output . It suffices to the norm

which is larger than a very large value ,i.e., forThen, a new relationship can be derived as shown in (26) at

the bottom of the page.Equation (26) indicates that even if and

are very small values (much less than 1) in per unit,may still be a

very large number compared with, which is the lower bound based on the conventional

model. Therefore, the effect of to cannot be ne-glected in this case. This means it can accelerate the systemfrequency deviation and make instability situation worse thanusing the conventional (17).

(26)


C. Largest Dip of Frequency Change

As shown in the previous analysis in Section III-B1, the fre-quency deviation in a stable case is bounded with the upperbound given by (24). Therefore, it is interesting to solve thelargest angular frequency dip [i.e., ] or itsLaplace transform [19]. This is because the largest fre-quency dip is one of the key specifications that power systemoperators want to know and compare against power systems sta-bility criterion. It can be obtained by Laplace inverse transfor-mation next.At the largest angular frequency dip , the partial

derivative of versus time must be zero, i.e.,

(27)

Here, because at the very beginning, there is noangular frequency deviation:

Thus (28)

Hence, we have equations for the following four cases:For SISO system without AGC case:

(29)

For SISO system with AGC case:

(30)

For multiple machines without AGC case:

(31)

For multiple machines with AGC case:

(32)

Solving (29)–(32) can give the time in both cases, when. If is an oscillation response, i.e., there

are several points such that , then choose thesmallest as since the first swing in a stable case givesthe largest frequency dip. Thus, the critical time and cor-responding largest dip can be derived respectively.Then, multiplication of , and with

can derive the largest dip of frequency change ,, and , respectively.

D. Proof of the Alignment of the Maximum Sensitivity and theLargest Frequency Dip

In a stable case, when or reachesits maximum, the necessary condition is that its derivative withtime should be equal to zero, i.e., .Then, applying Laplace transformation towith zero initial condition, i.e., and considering(3), (7), (11), or (15) which have the same formulation, we have

(33)

For a general time function , its Laplace transform isgiven by: . Thus, wehave

(34)

where and are defined as andin this

paper.In addition, we have and , which

can be derived by the definition of .Therefore, taking inverse Laplace transform on both sides of

(34), we have

(35)

To solve (33) to obtain the critical time , the inverseLaplace transformation is applied to (33). Also, (35) is used as


well. Here we consider the general initial condition ,which means no frequency deviation initially. Thus, we have

(36)

where is the impulse function with magnitude of 1. Thus,from (36), we have . Since

(i.e., no frequency deviation initially), therefore(36) indicates that solving is equiva-lent to solving .Hence, we can conclude that the maximum of is

aligned with the maximum of (i.e., the largest frequencydip). Therefore, the error of evaluating gives the largest im-pact to frequency deviation right at the time when the largestfrequency deviation occurs. This further shows the importanceof obtaining accurate the load-damping coefficient, . This im-portant feature can be easily observed in the simulation results.

IV. NUMERICAL SIMULATIONS

As listed below, eight case studies have been performed.• Single machine without AGC—Stable and Unstable Cases• Single machine with AGC—Stable and Unstable Cases• Multiple machines without AGC—Stable and UnstableCases

• Multiple machines with AGC—Stable and Unstable Cases

A. Single Machine (SISO) Without AGC

In this case the simulation time period is 20 seconds. Con-sider a typical aggregated power system containing a load anda single generator with a reheat turbine. Assume the system pa-rameters are , , ,

, , , . Here a load in-crease is considered as the external disturbance. Consider a stepload change, [4], which can be attributed toa demand response signal. In addition, let the be a set ofvalues with 20% increments: 20%, 40%, 60%, 80%, and 100%.The scenario of 100% increase of can be roughly viewed asthe extreme case that all loads are actually frequency sensitivewhile it is thought only half of the loads are sensitive. From allthe given parameters, it can be calculated that this power systemis stable. The Simulink diagram of a single machine case studywithout AGC is illustrated in Fig. 8.A.1) Single Machine Without AGC—Stable CaseFig. 9 shows the six curves: the “external disturbance” curve

is obtained using (17) by ignoring the impact of , while thefive “total disturbance” curves are obtained with (18) and (3) toaddress the impact from various values. If the six curves inFig. 9 are compared, they are very similar and close.Fig. 10 is derived from (29). The textboxes in Figs. 9 and 10

show the critical time at the largest frequency

Fig. 8. Simulink diagram without AGC.

Fig. 9. curves.

Fig. 10. curve with critical time.

dip, when Hz as shown in Fig. 9 or whenas shown in Fig. 10.

The sensitivity curve of frequency deviation, , w.r.t. load-damping coefficient, , is shown in Fig. 11. As shown in thefigure, the sensitivity function curve is relatively small, as op-posed to the case in unstable case shown later in this paper.An important observation in Fig. 11 is that

reaches its maximum also at the critical timewhen the maximum frequency dip occurs. This verifies theconclusion in the last paragraph in Section III-D that the error


Fig. 11. curve indicating that the maximum occurswhen the largest occurs.

Fig. 12. curve.

of gives the largest impact to frequency deviation right at thetime when the largest frequency deviation occurs. This furtherdemonstrates the importance of obtaining the accurate value.A.2) Single Machine Without AGC—Unstable CaseIn this case, we assume some disturbance happens to make

the generation governor unstable at the time . Inthis simulation, the parameter in the governor control in Fig. 8is changed from to , for demonstration pur-pose, to produce a pole in the right half of the plane. Here thesimulation time period is 1.4 sec, because the system frequencyis already close to the instability threshold (57 Hz [1], [2]) at1.4 sec. After that, the system frequency will sharply deviatefrom 60 Hz.The comparison result is shown in Fig. 12 with different

values. The sensitivity of the frequency deviation to theload-damping coefficient is shown in Fig. 13. These two fig-ures indicate that the power system is unstable and has a trend tobe even worse than anticipated using the conventional model in(17), and the load-damping coefficient can exert larger effecton SFR. So, it may accelerate the system frequency collapse inthis case. Therefore, the effect of load-damping coefficientcan be significant.

Fig. 13. curve.

Fig. 14. Simulink diagram with AGC.

Fig. 15. curve.

B. Single Machine (SISO) With AGC

In this subsection, the machine is equipped with AGC and theAGC parameter is 3. The Simulink diagram is as shown inFig. 14.B.1) Single Machine With AGC—Stable CaseSimilar to the results presented in Case A1, the results from

“external disturbance” based on (17) and “total disturbance”based on (18) are shown in Fig. 15.Fig. 16 is obtained from (30). The critical time, ,

and the largest frequency dip, , aregiven in the textboxes in Fig. 16 as well as Fig. 15, respectively.Similar to Case A1, the impact of the load-damping coef-

ficient on frequency in this case is also relatively small and



Fig. 17. curve indicating the maximum occurswhen the largest occurs.

bounded if compared with unstable cases. Note the presence ofAGC can recover the frequency back to the 60 Hz referencefrequency.Further, as shown in Fig. 17, reaches its max-

imum also at the critical time , when the largestfrequency dip occurs. Again, this verifies the conclusion in thelast paragraph in Section III-D and demonstrates the importanceof obtaining the accurate value.B.2) Single Machine With AGC—Unstable CaseSimilar to Case A2, here we assume a fault on the governor

starts at causes the parameter in Fig. 14 to change fromto for demonstration purpose. This leads

to a pole in the right half of the plane. In this case the simula-tion time duration is also 1.4 sec as in Case A2. The comparisonresult is shown in Fig. 18. The sensitivity of the frequency de-viation w.r.t. the load-damping coefficient is shown inFig. 19. The same observation and conclusion as the previouscase A2 can be drawn.

Fig. 18. curve.

Fig. 19. curve.

Fig. 20. Simulink diagram of multiple-machines case without AGC.

C. Multiple Machines Without AGC

In this and the next subsections, a two-machine system isconsidered. Let , ,

, , , ,, , , ,

, and . Further, consider a 10% load increase onthe first system input, i.e., [4]. Also considerthe actual is 20%, 40%, 60%, 80%, and 100%, respectively,higher than the expected value. The Simulink diagram is shownin Fig. 20.C.1) Multiple Machines Without AGC—Stable CaseIn this case the simulation time period is 20 sec. The six

curves obtained from (17) and (18) based on various


Fig. 21. curve.


values are shown in Fig. 21. The results of andare shown in Figs. 22 and 23, respectively. The

critical time, , and the largest frequencydip, , are given in the textboxes ofFigs. 21 and 22. Observations and conclusion are very similarto the ones in Cases A1 and B1. For example, the impact ofthe load-damping coefficient on SFR is relatively small andbounded if compared with unstable cases; reachesits maximum right at the critical time ; and theerror of gives the largest impact to frequency deviation rightat the time when the largest frequency deviation occurs.C.2) Multiple Machines Without AGC—Unstable CaseSimilar to Cases A2 and B2, here we assume a fault on Gov-

ernor 2 at causes the parameter to change fromto in Fig. 20 for demonstration purpose. This leadsto a pole in the right half of the plane. The simulation resultsare shown up to 2.4 sec. After that, the system frequency willsharply deviate from 60 Hz. The comparison result is shown inFig. 24. The sensitivity curve is shown in Fig. 25.


Fig. 24. curve.

Fig. 25. curve.

Similar to the A2 and B2 cases, Fig. 24 shows that thefrequency response using (18) is worse than the conventionalmodel using (17). Thus, the consideration of may lead to


Fig. 26. Simulink diagram multiple-machines case with AGC.

Fig. 27. curve.


acceleration of frequency instability and less response time forcorrective actions.

D. Multiple Machines With AGC

In this subsection, the system is equipped with AGC and theAGC parameter is 5. This is as shown in Fig. 26.D.1) Multiple Machines With AGC—Stable CaseFor this case, the six curves obtained from (17) and (18)

based on various values are shown in Fig. 27. The resultsof and are shown in Figs. 28 and29, respectively. The critical time, , and the


Fig. 30. curve.

largest frequency dip, , are given inthe textboxes of Figs. 27–29. Observations and conclusion arevery similar to the ones in Cases A1, B1 and C1. For example,the impact of the load-damping coefficient on SFR is alsorelatively small and bounded if compared with unstable cases;

reaches its maximum right at the critical time; and the error of gives the largest impact to

frequency deviation right at the time when the largest frequencydeviation occurs.Similar to Case B1, in this case AGC recovers the frequency

back to 60 Hz.D.2) Multiple Machines With AGC—Unstable CaseSimilar to Case C2, here we assume a fault on Governor 2 atcauses the parameter in Fig. 26 to change from

to for demonstration purpose. This leads to a polein the right half of the plane. Here the simulation results areshown up to 2.4 sec. The comparison result is in Fig. 30. The

sensitivity is shown in Fig. 31. The same observa-tion and conclusion as the previous unstable cases can be drawn.That is, the error in load-damping coefficient will lead to a


Fig. 31. curve.

frequency drop worse than expected such that the protection de-vice may trip quicker than anticipated.

V. CONCLUSIONS

The increasing penetration of controllable load calls the inter-ests to re-examine the load-frequency response. This paper in-vestigates the impact of the load-damping coefficient, becausemany load control programs target frequency-sensitive motorloads whichmay have a significantly different load-damping co-efficient, , from the rest of the loads.This paper investigates the effect of frequency-sensitive load

on the power system frequency regulation based on the typicalSFR model. Theoretic analyses as well as simulation studiesshow that the impact of an inaccurate load-damping coefficientis relatively small and bounded when the power system isessentially stable; while the system frequency deviation maybe accelerated when the power system is indeed unstable afterdisturbance.This paper also derives analytical calculation of the largest

frequency deviation and the corresponding critical time by in-verse Laplace transformation. This can be a useful indicator forpower system operators for decision-making of load control orinterruption.Further, this paper proves that the error of gives the largest

impact to frequency deviation right at the time when the largestfrequency deviation occurs. Simulation studies verify this con-clusion, which also demonstrates the importance of obtainingthe accurate value.The future research work may be about a strategic design of

a robust load shedding scheme considering the variation of theload-damping coefficient. Also, sensitivity study of other pa-rameters and other generator models such as hydraulic turbines,perhaps in a large-scale system, can be investigated.

REFERENCES[1] 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.

[2] P. M. Anderson and M. Mirheydar, “An adaptive method for settingunderfrequency load shedding relays,” IEEE Trans. Power Syst., vol.7, no. 2, pp. 647–655, May 1992.

[3] H. Saadat, Power System Analysis, 2nd ed. New York: McGraw-Hill,2002.

[4] P. Kundur, Power System Stability and Control. Palo Alto, CA:EPRI, 1994.

[5] S. Mukherjee, S. Teleke, and V. Bandaru, “Frequency response anddynamic power balancing in wind and solar generation,” in Proc. IEEEPES General Meeting, Jul. 2011.

[6] L. Yao and H. Lu, “A two-way direct control of central air-conditioningload via the internet,” IEEE Trans. Power Del., vol. 24, no. 1, pp.240–248, Jan. 2009.

[7] L. Yao, W. Chang, and R. Yen, “An iterative deepening genetic algo-rithm for scheduling of direct load control,” IEEE Trans. Power Syst.,vol. 20, no. 3, pp. 1414–1421, Aug. 2005.

[8] N. Ruiz, I. Cobelo, and J. Oyarzabal, “A direct load control model forvirtual power plant management,” IEEE Trans. Power Syst., vol. 24,no. 2, pp. 959–966, May 2009.

[9] V. V. Terzija, “Adaptive underfrequency load shedding based on themagnitude of the disturbance estimation,” IEEE Trans. Power Syst.,vol. 21, no. 3, pp. 1260–1266, Aug. 2006.

[10] Y. Halevi and D. Kottick, “Optimization of load shedding system,”IEEE Trans. Power Convers., vol. 8, no. 2, pp. 207–213, Jun. 1993.

[11] X. Lin, H. Weng, Q. Zou, and P. Liu, “The frequency closed-loop con-trol strategy of islanded power systems,” IEEE Trans. Power Syst., vol.23, no. 2, pp. 796–803, May 2008.

[12] D. L. H. Aik, “A general-order system frequency response model in-corporating load shedding: Analytic modeling and applications,” IEEETrans. Power Syst., vol. 21, no. 2, pp. 709–717, May 2006.

[13] N. Amjady and F. Fallahi, “Determination of frequency stability borderof power system to set the thresholds of under frequency load sheddingrelays,” Energy Conver. Manage., vol. 51, pp. 1864–1872, 2010.

[14] H. Bevrani, G. Ledwich, Z. Dong, and J. J. Ford, “Regional frequencyresponse analysis under normal and emergency conditions,” Elect.Power Syst. Res., vol. 79, pp. 837–845, 2009.

[15] H. Omara and F. Bouffard, “A methodology to study the impact of anincreasingly nonconventional load mix on primary frequency control,”in Proc. IEEE PES General Meeting, Jul. 2009.

[16] N. Lu, D. P. Chassin, and S. E. Widergren, “Modeling uncertaintiesin aggregated thermostatically controlled loads using a state queueingmodel,” IEEE Trans. Power Syst., vol. 20, no. 2, pp. 725–733, May2005.

[17] A. Karakas, F. Li, and S. Adhikari, “Aggregation of multiple inductionmotors using MATLAP-based software package,” in Proc. IEEE PESPower System Conf. and Expo. (PSCE) 2009, Seattle, WA,Mar. 15–18,2009.

[18] F. Nozari, M. D. Kankam, and W.W. Price, “Aggregation of inductionmotors for transient stability load modeling,” IEEE Trans. Power Syst.,vol. 2, no. 4, pp. 1096–1103, Nov. 1987.

[19] I. Egido, F. Fernandez-Bernal, P. Centeno, and L. Rouco, “Maximumfrequency deviation calculation in small isolated power systems,”IEEE Trans. Power Systems, vol. 24, no. 4, pp. 1731–1738, Nov. 2009.

Hao Huang (S’09) is pursuing the Ph.D. degree at the University of Tennessee(UT), Knoxville.His major research interest is controllable load under market operations.

Fangxing (Fran) Li (M’01–SM’05) received the Ph.D. degree from VirginiaTech, Blacksburg, in 2001.He is presently an Associate Professor at the University of Tennessee (UT),

Knoxville. Prior to joining UT in 2005, he was a principal R&D engineer atABB Electrical System Consulting (ESC).Dr. Li is a registered Professional Engineer (PE) in the state of North Carolina

and a Fellow of IET.

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1324 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL....

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