Further Results on Delay-Dependent Stability of Multi-Area Load ...

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013

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Further Results on Delay-Dependent Stability of Multi-Area Load Frequency Control Chuan-Ke Zhang, Student Member, IEEE, L. Jiang, Member, IEEE, Q. H. Wu, Fellow, IEEE, Yong He, Senior Member, IEEE, and Min Wu, Senior Member, IEEE

Abstract—Further to results reported by Jiang et al., this paper investigates delay-dependent stability of load frequency control (LFC) emphasizing on multi-area and deregulated environment. Based on Lyapunov theory and the linear matrix inequality technique, a new stability criterion is proposed to improve calculation accuracy and to reduce computation time, which makes it be suitable for handling with multi-area LFC schemes. The interaction of delay margins between different control areas and the relationship between delay margins and control gains are investigated in details. Moreover, usage of delay margins as a new performance index to guide controller design is discussed, including tuning of the controller for a trade-off between delay tolerance and dynamic response, choosing the upper bound of the fault counter of communication channels and the upper bound of sampling period of a discrete realization of the controller. Case studies are carried out based on two-area traditional, two-area and three-area deregulated LFC schemes, all equipped with PID-type controllers, respectively. Simulation studies are given to verify the effectiveness of the proposed method. Index Terms—Communication delays, delay margin, delay-dependent stability, deregulated environment, multi-area load frequency control.

I. INTRODUCTION

T

RADITIONAL load frequency control (LFC) employs a dedicated communication channel to transmit measurement and control signals, while the LFC under deregulated environment, such as bilateral contract between generation companies (Gencos) and distribution companies (Discos) for the provision of load following and third-party LFC service, tends to use open communication networks [1]–[4]. Although the constant

Manuscript received December 11, 2012; revised April 10, 2013; accepted May 17, 2013. Date of publication June 14, 2013; date of current version October 17, 2013. This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 61210011 and 61125301. Paper no. TPWRS-01353-2012. C.-K. Zhang is with the Hunan Engineering Laboratory for Advanced Control and Intelligent Automation, School of Information Science and Engineering, Central South University, Changsha 410083, China, and also with the Department of Electrical Engineering & Electronics, The University of Liverpool, Liverpool, L69 3GJ, U.K. (e-mail: [email protected]). L. Jiang and Q. H. Wu are with the Department of Electrical Engineering & Electronics, The University of Liverpool, Liverpool, L69 3GJ, U.K. (e-mail: [email protected]; [email protected]). Y. He and M. Wu are with the Hunan Engineering Laboratory for Advanced Control and Intelligent Automation, School of Information Science and Engineering, Central South University, Changsha 410083, China (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.2013.2265104

delay introduced by the dedicated communication channel is normally ignored, the breakdown of the communication channel itself could be converted to time delays [9]. With the introduction of open communication channel, constant and time-varying delays will be introduced in the LFC scheme due to packet dropout and disordering [4]–[6], updating of area control error (ACE) signal [7], and faults of communication channels [9]. The delays induced by the communication channels will degrade the dynamic performance and may even cause instability of an LFC scheme [4], [8]–[10]. The instability of the LFC scheme means that the ACE and frequency deviation will deviate far away from zero, which makes the control areas impossibly comply with the control performance standards, CPS1 and CPS2, adopted by North American Electric Reliability Council (NERC) [22]. Analysis/synthesis of the LFC in the presence of communication delays has been addressed, such as robust controllers based on linear matrix inequality (LMI) technique [11] and robust detheory [12], [13]. Rocentralized PI-type LFC based on bustness of those controllers against time delays was normally verified by simulation studies only. Recently, authors of this paper [9] investigated the delay-dependent stability of the LFC via calculating delay margins and investigating the relationship between the delay margins and controller parameters. The investigation of [9] focuses on traditional LFC schemes, while the LFC schemes in deregulated environment, which more commonly encounters time delays due to the usage of open communication networks, as well as how to use the delay margins as a performance index to guide the design and operation of the LFC controller, have not been investigated. Moreover, the stability criterion used in [9] is also deserved to be improved from the following two aspects. The first one is that it contains too many decision variables such that the total computation time increases sharply with the increasing of the problem dimension, which results in the difficulty of applying the proposed method to deal with multi-area LFC schemes. Secondly, the conservativeness of the stability criterion used cannot reveal the coupling dynamic between different control areas which results that the multiple delays of different control areas have been found be independent, i.e., looking like a rectangle or cube in [9]. On the other hand, the practical LFC controllers are operated in discrete mode as the ACE signals of the LFC scheme are usually updated in a period from 2 s–4 s [7]. It is found in [14] that the optimal integral controller gains designed in the continuous mode cannot be applied directly in discrete mode, while the simulation studies in [7] revealed that a relatively large sampling period around 20 s can still result in satisfactory results for some special cases. To the authors’ best knowledge, how to

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select the sampling period and determine the acceptable upper bound of the sampling period (UBSP) for a given LFC controller have not been investigated analytically yet. In fact, a continuous controller with an input delay can be used to model such sample-based controller, in which the input delay is bounded by the sampling period [15]. Based on this understanding, the delay margin can be used as the UBSP to guide the choice of the sampling period for an LFC controller. Based on the aforementioned discussions, this paper investigates the delay-dependent stability of a multi-area LFC in deregulated environment and tackles the unsolved problems of [9]. An improved stability criterion with less conservativeness and faster calculation speed is derived by using Jensen inequality [16], compared with the criterion used in [9]. The proposed method can reveal the interaction of delay margins for different control areas properly, and it is more suitable to deal with the multi-area LFC schemes. How to use delay margins as a performance index to guide the design and operation of the LFC schemes is also discussed, including using delay margins to set the upper bound of the fault counter (UBFC) to extend the on-service period of the LFC scheme when the communication channel is broken down, as discussed in [9], and set the UBSP for a discrete mode LFC controller; and defining the delay margin as a new performance index to tune the LFC control gains. The remainder of the paper is organized as follows. Section II gives the dynamic models of both traditional and deregulated multi-area LFC schemes, including delays. Section III presents an improved stability criterion and steps of the proposed method. In Section IV, case studies are based on deregulated two-area and three-area LFC schemes, and traditional two-area LFC scheme as well. Effectiveness of the proposed method is verified by simulation studies and the application of delay margins to guide the synthesis of the LFC controller is also shown. Conclusions are given in Section V. II. DYNAMIC MODEL OF LFC SCHEME WITH DELAYS

A. Deregulated Multi-Area LFC Scheme For multi-area LFC in deregulated environment, as shown in Fig. 1 including the dotted line connection, in which each Genco can contract with various Discos in or out of the area this Genco belonging to. Those bilateral contracts are usually visualized by an augmented generation participation matrix (AGPM). For a large-scale power system with areas and Discos and Gencos in each area, the AGPM is given by ..

.

.. .

where

.. .

..

.

.. .

with , and shows the participation factor of Genco in the total load following requirement of Disco based on the possible contracts. The dotted lines in Fig. 1 show interfaces between areas and the demand signals based on the possible contracts. The followings are obtained [17]: (2) (3)

(4)

This section describes the model of the LFC in deregulated environment, which includes the traditional LFC scheme as a special case. The structure of the th control area, including generating units, is shown in Fig. 1. To simplify the analysis, the delays, including communication delays, sample-induced delays, and fault-induced delays, can be combined as one single delay and represented by an exponential block , as shown in Fig. 1.

.. .

Fig. 1. LFC structure of control area (Traditional LFC: without dotted line connection; Deregulated LFC: with dotted line connection).

(5) (6) (7) and are the total contracted and un-conwhere tracted demands in area , respectively; and the contracted and un-contracted demands of the th Disco in area , respectively; the scheduled power tie line power flow between areas and ; and the desired total power generation of the th Genco in area . The state-space model for area can be obtained as

(1) (8)

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where

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From (10), there exists coupling between states of area and area . Redefine the state vector be , then the whole closed-loop system is rewritten as (11) where .. .

..

.

.. .

B. Traditional Multi-Area LFC Scheme Model of a traditional LFC system can be obtained by excluding the dotted line connection of Fig. 1, as shown in the following: (12) has similar structure of in (11) and can be obwhere tained by re-defining the and be and are the deviation of frequency, tie-line power exchange, generator mechanical output, and valve position, respectively; the moment of inertia of generator unit, generator unit damping coefficient, time constant of the governor, time constant of the turbine, and speed drop, respectively; the frequency bias factor; the ramp rate factor; and the ACE. For area , using as corresponding control input, a PID controller is designed as follows:

(9) where , and are proportional, integral, and differential gains, respectively, define . To simplify the analysis, the PID control problem is transformed into a static output feedback control problem [18]. Define the following virtual state and output vectors and , the closed-loop system can be rewritten as

The difference between those two models is their disturbance components. C. Objective of the Paper This paper aims at two issues: 1) proposing an improved stability criterion with higher accuracy and less computation time to determine delay margins of multi-area LFC schemes and to reveal the interaction effects between different areas; 2) showing how to use delay margins to guild the tuning of the controller gains and the choice of two key parameters, UBSP and UBFC, of a known controller. III. DELAY-DEPENDENT STABILITY ANALYSIS METHOD In this section, an improved delay-dependent criterion is derived by using Jensen inequality [16] to remedy the drawbacks of the criterion used in [9]. Delay margins are calculated by using the LMI technique. A. Improved Stability Criterion

(10) where

Theorem 1: Consider the following time-delay system: (13) For given scalars satisfying , the system is asymptotically stable if there exist matrices , , and , , such that (14)

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where

Step 3) Draw the trajectory of delay margins based on calculated , and then the region surrounded by the trajectory and coordinate axis is the stability region. Step 4) Verify the accuracy of the calculated value via simulation method based on the detailed model of the LFC scheme considering the physical constraint. IV. CASE STUDIES

Proof: Construct a candidate Lyapunov functional as

(15) where , , and which means

are positive define symmetric matrices, . It follows from Jensen inequality that (16)

, then calculating the where derivative of (15) and applying (14) and (16) yield with . Therefore, the system is stable if , , and and (14) holds. Compared with the criterion used in [9], Theorem 1 reduces conservativeness by taking into account relationship between different delays during the construction of Lyapunov functional, as shown in the second and third terms of . Moreover, the total number of decision variables for the criteria used in [9] and in this paper is respectively given as

(17) where is the number of delays (or control areas) and is the dimension of the system, which shows a great reduction of decision variables with the total number of control areas. B. Analysis Steps of the Proposed Method Take a two-area LFC scheme as example, the whole analysis process of the proposed method can be summarized as follows: Step 1) Calculate the state-space model of the closed-loop LFC equipped with a PID controller, as shown in Section II. Step 2) The trajectory of delay margins of two-area LFC scheme can be described by a set of polar coordi[9]. Calcunate points with with respect to by combining the late solver in MATLAB and the binary search algorithm, as reported in [9].

Case studies are based on a traditional two-area LFC scheme and deregulated two/three-area LFC schemes, respectively. Each area of the traditional two-area LFC scheme and the deregulated three-area LFC scheme is assumed to contain one Genco and one Disco. Each area of the deregulated two-area power system includes two Gencos and two Discos. The related parameters given in [9] and [17] are used and listed in the Appendix. Controller gains of each control area are assumed to be equal and time delays of each control area are treated as different. Delay margins are calculated based on the proposed method by using the MATLAB/LMI toolbox [19] and verified by simulation studies of the closed-loop system. Then the comparison of calculation time spent by different methods and how to guide the controller design via the calculated delay margins are discussed. A. Delay Margin Calculation 1) Traditional Two-Area LFC: Delay margins of a two-area LFC equipped with an I-controller , a PI-controller , , or a PID-controller ( , , ) are respectively calculated. The stability region is shown in Fig. 2, compared with the results obtained in [9]. Some typical values are listed in Table I, where , with and respectively being delay margins of two areas. 2) Deregulated Two-Area LFC: Delay margins of a two-area ,a LFC equipped with an I-controller , ), or a PID-conPI-controller ( ) are troller ( , , respectively calculated. The stability region is shown in Fig. 3, compared with the one obtained by the method used in [9]. Typical values are listed in Table II. 3) Deregulated Three-Area LFC: Delay margins of a three, a PI-conarea LFC equipped with an I-controller , , ), or a PID-controller ( troller ( , ) are respectively calculated. The stability region is shown in Fig. 4. Some typical values are listed and angles and in Tables III–V, in which the magnitude are defined as , , in the with being the projection of space, respectively. 4) Observations and Discussions: Since Theorem 1 only provides sufficient conditions, there exists conservativeness of the estimated delay margin. The resulting stability regions shown in Figs. 2–4 are smaller than their real regions. However, such conservatives will not affect the trends of relationship between delay margins and control parameters. Moreover, the less conservative conditions developed and used in this paper

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Fig. 2. Stability regions of the traditional two-area LFC obtained by the methods in this paper [TP] and [9]. (a) I-control. (b) PI-control. (c) PID-control.

Fig. 3. Stability regions of the deregulated two-area LFC obtained by the methods in this paper [TP] and [9]. (a) I-control. (b) PI-control. (c) PID-control.

DELAY MARGIN

TABLE I (TRADITIONAL TWO-AREA LFC)

DELAY MARGIN

TABLE II (DEREGULATED TWO-AREA LFC)

would provide a more accurate observation for such a relationship. Specifically, the following observations are summarized based on the obtained results. • For both traditional and deregulated LFC schemes, the stability region becomes smaller with the increasing of and ; the stability region is nearly in direct proportion to , as shown in Figs. 2(a) and 3(a); and the change

of the region with respect to is more complex, for the two-area LFC, the region decreases as the increasing of , as shown in Figs. 2(c) and 3(c), while for the three-area LFC, it increases a little as the increasing of , as shown in Tables IV and IV . • The results obtained by the proposed method show that the coupling dynamic between different areas affect their delay margins for both traditional and deregulated LFC schemes. Unlike the regular stability region obtained in [9], the shape of stability regions obtained in this paper is irregular, which reveals the fact that the delay margins of different areas are affected by each other. For most of control gains (except for the deregulated LFC with or ), as the time delay of one area increasing, the delay margin of the other area increases firstly and then decreases; and the delay margin of area 1 when area 2 has no delay is smaller than the one when area 2 has a delay, for example, Fig. 2(a) shows that the delay margin of area 2 is 13.7 s when and it increases to 14.5 s when . This observation shows that the delay in one area may increase the delay margin of the other area. • The comparison of Figs. 2 and 3 shows that the delay margins of the deregulated LFC scheme have closer interaction than those of the traditional LFC scheme. For the traditional LFC, when the delay of one area varies from 0 to its margin, the variation of the delay margin of the other area is not obvious. While for the deregulated LFC, the delay

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Fig. 4. Stability regions of the deregulated three-area LFC obtained by the methods in this paper. (a) I-control. (b) PI-control. (c) PID-control.

DELAY MARGIN

TABLE III (DEREGULATED THREE-AREA LFC, I-CONTROL)

DELAY MARGIN

TABLE IV (DEREGULATED THREE-AREA LFC, PI-CONTROL)

DELAY MARGIN

TABLE V (DEREGULATED THREE-AREA LFC, PID-CONTROL)

the stability regions obtained via the proposed method are larger than the ones obtained by the method used in [9], which is achieved by reducing the conservativeness of the criterion. Secondly, the proposed method has revealed the interaction of different areas more accurately. For traditional LFC, the method used in [9] cannot find the interaction for different areas since the coupling connections are not obvious, while the proposed method can reveal more details of such interaction. For deregulated LFC, the method of [9] shows that, when delay of one control area increases, the delay margin of the other control area reduces. However, based on the results from the proposed method, for some special control parameters, an increment of delay (especially from 0) in one control area may lead to an increasing of delay margin of the other area. The similar trend for a simple time-delay system with two delays has been found by the accurately theoretical analysis in [23]. This observation will also be further discussed using simulation studies. 5) Simulation Verification: Simulation studies are carried out to verify the calculation accuracy of the proposed method. The results of the two-area deregulated LFC system equipped with a PI controller ( , ) and the angle are given only due to page limits. The generator rate constraint (GRC) is assumed to be 0.1 pu/min [17], and the update period of ACE signals is 2 s [7]. All Discos contract with the available Cencos as the following AGPM: (18)

margin of one area changes obviously with the variation of the delay in the other area. It is worth pointing out that such difference is caused by the different system parameters, such as , , , , , etc., but not by the deregulated environment, since the deregulation only complicates the load disturbances, as shown in (11) and (12), and it has no directly effect on the internal stability of control systems [21]. • Based on the comparison of the results obtained by the methods proposed in this paper and used in [9], two aspects of advantages for the proposed method are found. Firstly,

Assume that a step load of 0.1 pu is demanded by each Disco in the areas, and Disco 1 in area 1 and Disco 2 in area 2 all demand 0.08 pu as large un-contracted loads, i.e., (19) Test the performance of the closed-loop LFC scheme by increasing the time delay from 0 step-by-step till the system becomes unstable. The responses of area 1 for different delays are shown in Fig. 5, the results show that the magnitude of two delay margins, , is within the range [10 s, 12 s]. The result obtained

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TABLE VI FOR THE PROPOSED METHOD, CALCULATION TIME: FOR THE METHOD USED IN [9] (TRADITIONAL TWO-AREA LFC)

B. Calculation Time Comparison

Fig. 5. Responses of area 1 for the PI-based deregulated LFC scheme ( , ) with different delays.

Fig. 6. Responses of area 1 for the PI-based deregulated LFC scheme ( , ) with different delays.

This subsection describes the improvement of calculation speed. Based on a same PC with a Intel i5 CPU and a memory of 4 GB, the average calculation time spent by the proposed method for the traditional two-area LFC equipped with different controllers is found to be about 10 s, and the average calculation time spent by the method used in [9] is found to be about 2500 s. Those two average values are obtained based on the 63 groups of calculation time for different parameters (control gain and the angle shown in Table I). Parts of the calculation time for the LFC equipped with a PI controller ( , ) are listed in Table VI. The required time of the proposed method is about only 0.4% of the one of the published method. The reason is that the number of decision variables of the criterion has been reduced dramatically from 1863 to 225, based on (17). Similarly, the average calculation time spent by the methods proposed in this paper are about 70 s and 350 s for the deregulated two and three-area LFC schemes, respectively, while those for the method used in [9] are larger than and , respectively. The proposed method has greatly reduce the calculation time, which is mainly benefited from the reducing of the number of decision variables: from 3861 to 455 for the deregulated two-area LFC and from 13 230 to 735 for the deregulated three-area LFC, based on (17). The proposed method has greatly reduced the time spent on the delay margin calculation, which makes it be more suitable to deal with the multi-area LFC schemes. C. Guide the Controller Design Via Delay Margins

is 9.73 s by the method used in [9] and is 10.57 s by the proposed method, which shows a better accuracy. Moreover, simulation studies are also used to verify the claim in Section IV-A4) that the increment of (especially from 0) delay in one area may increase the delay margin of the other area for special control parameters. Taking the two-area deregulated LFC system equipped with a PI controller ( , ) as an example, the responses of area 1 for two cases (Case I: and and Case II: and ) are shown in Fig. 6. From the figure, the system is unstable for Case I since the is larger than its delay margin 9.88 s (listed in Table II); while, for Case II, the system becomes stable since the delay margin of area 1 increases with the delay of area 2, , increasing from 0 to 3.5 s. In other words, the obtained results shown in Figs. 2–4 could reveal more details of the interaction of delay margins between different control areas.

As mentioned in [9] and [18], delay margins can be used as an additional performance index to guide the synthesis of the LFC controllers. This subsection demonstrates how to use them to determine two key parameters of an LFC controller, UBFC and UBSP, and as a new performance index to tune the controller gains. 1) Upper Bound of the Communication Fault Counter (UBFC): As mentioned in Section I, the delay margin can be used to determine a relatively bigger UBFC to extend the on-service time of the LFC when a communication fault happens, compared with a conservative and smaller UDFC chose based on operational experience. The deregulated two-area LFC equipped with an I controller is taken as an example. The operating conditions, including AGPM, GRC, update period of ACE, and demanded loads, are assumed same as the ones defined in Section IV-A5). From Table II and Fig. 3(a), the system is stable when time delays of two areas

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Fig. 7. Responses of area 1 for the I-based deregulated LFC scheme with different fault cases.

are less than 14 s. The responses of the following three cases are shown in Fig. 7. • Case I: normal condition without fault. • Case II: a fault occurs at 16 s and is cleared at 30 s, and the UBFC is set to be 14 s (7 periods) based on calculated delay margin. From 16 s to 30 s, ACE signal is no longer updated and remains constant by a zero-order holder, and control signal is calculated using the last updated ACE. • Case III: a fault occurs at 16 s and is clear at 30 s, and the UBFC is set to be 6 s (3 periods) based on operational experience. The controller stops at 22 s, and restarts when the fault is cleared at 30 s. The results show that the performance of Case II is better than that of Case III ( and ) or is similar to that of Case III . Thus, a larger UBFC can be set to improve the performance of the LFC under a communication channel fault. Moreover, for Case II, the controller does not need to be stopped and restarted since the ACE renews before the fault duration reaching the preset UBFC of 14 s. Thus, if the unavailability period of the communication channel due to fault is less than the UBFC, the LFC controller need not to be stopped so as to increase the on-service period of the LFC. 2) Upper Bound of Sampling Period (UBSP): The delay margin can be used to set the UBSP for a practical LFC. The deregulated two-area LFC system equipped with I-controller is tested under the same AGPM, GRC, and demanded loads as Section IV-A5). The responses under different sampling periods are shown in Fig. 8. The system is stable when the sampling period is less than the delay margin 14 s. Moreover, the results also show that the performance has not been degraded very much when using a bigger sampling period. Thus, for a practical LFC scheme, a larger sampling period can be set based on the delay margin calculated, which can reduce the actuation effort of the controller and may improve the longevity of the actuator such as the steam valve, and also reduce the bandwidth requirement of the communication channel. As in deregulated power systems, cost minimization has become the primary goal

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Fig. 8. Responses of area 1 for the I-based deregulated LFC scheme with different periods of ACE updating.

of all parties in the power market, thus a larger sampling period may help to reduce the operational and maintenance costs. 3) Tuning of Controller: The delay margin can be used as an additional performance index to tune controller based on trade-off between delay margin and dynamic performance [18] or combine with other tuning method [20]. The one-area LFC discussed in [20] is taken as an example. The tuning of the PID controller are dependent on two parameters and , which are artificially selected in [20]. For , and , , gains obtained using the method proposed in [20] are (20) (21) The delay margins provided by and are calculated as 0.56 s and 7.94 s. The frequency response of the system undergoing a positive load disturbance of 0.01 pu is shown in Fig. 9. When there is no delay, although provide a better transient response than , transient dynamic provided by is still acceptable; when a small delay is applied, the performance under is better than ; when the delay increases to 5 s, larger than the delay margin provided by , the system equipped with loses the stability while still maintains the stability of the system. Considering the inevitable time delay introduced by the communication networks, is better because of its great improvement of robustness against delays. D. Remark The stability criterion proposed is only used to determine the asymptotical stability of the LFC, i.e., the ACE and frequency deviation caused by disturbances converges back to zero after a limited period. Since the detailed transient response of the LFC, such as settling time and overshoot of the ACE, etc., has not been considered, the compliance with the control performance standards CPS1/CPS2 cannot be ensured by the stability criterion proposed. However, it is worth pointing out that the asymptotical stability of the LFC is the necessary condition for guaranteeing the control area to comply with the CPS1/CPS2 stan-

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TABLE VII TRADITIONAL TWO-AREA LFC SCHEME

TABLE VIII DEREGULATED TWO-AREA LFC SYSTEM

TABLE IX DEREGULATED THREE-AREA LFC SYSTEM Fig. 9. Frequency deviation responses of the system with respect to different . (c) . time delays. (a) without delay. (b)

dards. Furthermore, the impact of the time delay on those standards and the link between the time delay and those standards are worth investigating, especially for the LFC under deregulated electric markets and using open communication networks.

V. CONCLUSIONS The delay-dependent stability of the multi-area LFC schemes in deregulated environment has been investigated. The deregulated LFC scheme installed with PID-type controllers has been modeled as a linear system with multiple delays, including the traditional LFC scheme as a special case. To deal with the increased problem dimension caused by multi-area LFC schemes and reveal the interactions between different control areas, an improved LMI-based delay-dependent stability criterion, which has less conservatism and fewer decision variables than the existing criteria, has been derived to calculate the delay margins. The results have shown that the proposed method leads to an enlarged stability region, improved calculation accuracy, reduced computation time, and better visibility of the interaction for the delay margins of different areas. The case studies have been carried out based on two-area traditional and two/three-area deregulated LFC schemes. It has been found that delay margin decreases with the increasing of the proportional or integral gain of PID-type controllers for a fixed derivative gain, while for fixed proportional and integral gains, it decreases (for two-area LFC) or increases (for threearea LFC) with the increasing of the derivative gain. Moreover, the interaction between delay margins of different areas is more obvious for the deregulated LFC scheme. Applications of the delay margins to choose the UBFC for communication channels, the UBSP, and to tune the controller gains have shown the effectiveness of usage of the delay margin as a new performance index to design an LFC controller considering the impacts of communication networks.

APPENDIX Table VII has the traditional two-area LFC scheme. Table VIII has the deregulated two-area LFC system. Table IX has the deregulated three-area LFC system. REFERENCES [1] H. Bevrani, Robust Power System Frequency Control. New York, NY, USA: Springer, 2009. [2] V. Donde, M. A. Pai, and I. A. Hiskens, “Simulation and optimation in an AGC system after deregulation,” IEEE Trans. Power Syst., vol. 16, no. 3, pp. 481–489, Aug. 2001. [3] K. Tomsovic, D. E. Bakken, V. Venkatasubramanian, and A. Bose, “Designing the next generation of real-time control, communication, and computations for large power systems,” Proc. IEEE, vol. 93, no. 5, pp. 965–979, May 2005. [4] S. Bhowmik, K. Tomsovic, and A. Bose, “Communication models for third party load frequency control,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 543–548, Feb. 2004. [5] W. Yao, L. Jiang, Q. H. Wu, J. Y. Wen, and S. J. Cheng, “Delay-dependent stability analysis of the power system with a wide-area damping controller embedded,” IEEE Trans. Power Syst., vol. 29, no. 1, pp. 233–240, Feb. 2011. [6] S. Wang, X. Meng, and T. Chen, “Wide-area control of power system through delayed network communication,” IEEE Trans. Control Syst. Technol., vol. 20, no. 2, pp. 495–503, Mar. 2012. [7] J. Nanda, A. Mangla, and S. Suri, “Some new findings on automatic generation control of an interconnected hydrothermal system with conventional controllers,” IEEE Trans. Energy Convers., vol. 21, no. 1, pp. 187–184, Mar. 2006. [8] M. Wu, Y. He, and J. H. She, Stability Analysis and Robust Control of Time-Delay System. New York, NY, USA: Spring-Verlag, 2010. [9] L. Jiang, W. Yao, J. Y. Wen, S. J. Cheng, and Q. H. Wu, “Delay-dependent stability for load frequency control with constant and time-varying delays,” IEEE Trans. Power Syst., vol. 27, no. 2, pp. 932–941, May 2012. [10] Ibraheem, P. Kumar, and D. P. Kothari, “Recent philosophies of automatic generation control strategies in power systems,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 346–357, Feb. 2005. [11] X. Yu and K. Tomsovic, “Application of linear matrix inequalities for load frequency control with communication delays,” IEEE Trans. Power Syst., vol. 19, no. 3, pp. 1508–1515, Aug. 2004.

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[12] H. Bevrani and T. Hiyama, “Robust decentralised PI based LFC design for time delay power systems,” Energy Convers. Manage., vol. 49, no. 2, pp. 193–204, Feb. 2008. [13] H. Bevrani and T. Hiyama, “On load-frequency regulation with time delays: Design and real-time implementation,” IEEE Trans. Energy Convers., vol. 24, no. 1, pp. 292–300, Mar. 2009. [14] J. Nanda, M. L. Kothari, and P. S. Satsangi, “Automatic generation control of an interconnected hydrothermal system in continuous and discrete modes considering generation rate constraints,” IEE Proc. D Control Theory Applicat., vol. 130, no. 1, pp. 17–27, Jan. 1983. [15] E. Fridman, A. Seuret, and J. P. Richard, “Robust sampled-data stabilization of linear systems: An input delay approach,” Automatica, vol. 40, no. 8, pp. 1441–1446, Aug. 2004. [16] K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems. Boston, MA, USA: Birkhäuser, 2003. [17] H. Shayeghi, H. A. Shayanfar, and A. Jalili, “Multi-stage fuzzy PID power system automatic generation controller in deregulated environments,” Energy Convers. Manage., vol. 47, no. 18–19, pp. 2829–2845, Nov. 2006. [18] C. K. Zhang, L. Jiang, Q. H. Wu, Y. He, and M. Wu, “Delay-dependent robust load frequency control for time delay power systems,” IEEE Trans. Power Syst., to be published. [19] G. Balas, R. Chiang, A. Packard, and M. Safonov, Robust Control Toolbox User’s Guide. Natick, MA, USA: MathWorks, 2010. [20] W. Tan, “Unified tuning of PID load frequency controller for power systems via IMC,” IEEE Trans. Power Syst., vol. 25, no. 1, pp. 341–350, Feb. 2010. [21] W. Tan, H. Zhang, and M. Yu, “Decentralized load frequency control in deregulated environments,” Int. J. Elect. Power Energy Syst., vol. 41, no. 1, pp. 16–26, Oct. 2012. [22] N. Jaleeli and L. S. VanSlyck, “NERC’s new control performance standards,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 1092–1099, Aug. 1999. [23] N. Olgac and R. Sipahi, “Complete stability robustness of third order LTI multiple time delay systems,” Automatica, vol. 41, no. 8, pp. 1413–1422, Aug. 2005.

Chuan-Ke Zhang (S’12) received the B.Sc. degree in automation from Central South University, China, in 2007. He is currently pursuing the Ph.D. degree in Central South University. He was working as a visiting research student in the Department of Electrical Engineering and Electronics, The University of Liverpool, Liverpool, U.K.,

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013

from 2011 to 2013. His current research interests are time-delay systems, chaos synchronization, and power system stability and control.

L. Jiang (M’00) received the B.Sc. and M.Sc. degrees in electrical engineering from Huazhong University of Science and Technology (HUST), China, in 1992 and 1996, respectively, and the Ph.D. degree from the University of Liverpool, Liverpool, U.K., in 2001. He is a Lecturer in The University of Liverpool. His current research interests are control and analysis of power system, smart grid and renewable energy.

Q. H. Wu (M’91–SM’97–F’11) received the M.Sc. degree in electrical engineering from Huazhong University of Science and Technology (HUST), China, in 1981 and the Ph.D. degree in electrical engineering from The Queen’s University of Belfast (QUB), U.K., in 1987. He has been the Chair of Electrical Engineering in the Department of Electrical Engineering and Electronics, The University of Liverpool, Liverpool, U.K., since September 1995. His current research interests are advanced control techniques, computational intelligence, and power and energy systems.

Yong He (SM’06) received the B.Sc. and M.Sc. degrees in applied mathematics from Central South University (CSU), China, in 1991 and 1994, respectively, and the Ph.D. degree in control theory and control engineering from CSU in 2004. He joined the staff of CSU in 1994, and is currently a Professor in the School of Information Science and Engineering, CSU. His current research interests are time-delay systems, networked control systems, and robust control.

Min Wu (SM’08) received the B.Sc. and M.Sc. degrees in engineering from Central South University (CSU), China, in 1983 and 1986, respectively, and the Ph.D. degree in engineering from the Tokyo Institute of Technology, Japan, in 1999. He joined the staff of CSU in 1986, and is currently a Professor in the School of Information Science and Engineering, CSU. His current research interests are robust control and its applications, process control, and intelligent control.