Gain and Phase Margins Based Delay Dependent Stability Analysis of Two- Area LFC System with Communication Delays

Transcription

1 Gain and Phae Margin Baed Delay Dependent Stability Analyi of Two- Area LFC Sytem with Communication Delay Şahin Sönmez and Saffet Ayaun Department of Electrical Engineering, Niğde Ömer Halidemir Univerity, Main Campu, 514, Niğde Turkey Abtract Thi paper invetigate the effect gain and phae margin () on delay-dependent tability analyi of a two-area load frequency control (LFC) ytem with communication delay. An frequency-domain baed exact method that take into account both gain and phae margin i utilized to determine tability delay margin in term of ytem and controller parameter. A gain-phae margin teter (T) i introduced to the LFC ytem a to take into gain and phae margin in delay margin computation. For a wide range of proportional integral (PI) controller gain, time delay value at which LFC ytem i both table and have deired tability margin meaured by gain and phae margin are computed. The time-domain imulation tudie indicate that delay margin mut be determined conidering gain and phae margin to have a better dynamic performance in term of fat damping of ocillation, le overhoot and ettling time. 1. Introduction Time delay have become an important iue in LFC ytem ince they degrade the performance of controller and adverely affect ytem dynamic and tability [1-3]. Time delay are experienced in LFC ytem becaue of the ue of an open and ditributed communication network ued to tranmit data from power plant to central controller delay [4]. With the ue of open communication infratructure, large amount of communication delay in the range of 5-15 are generally oberved in LFC ytem [, 6]. Such delay hould not be ignored and mut be taken into account in the tability analyi and controller deign of LFC ytem. The exiting tudie in the tability analyi of time-delayed LFC ytem mainly focu on the tability delay margin computation for a given et of controller parameter. Delay margin computation method could be grouped into two main type, namely frequency-domain direct method [3, 6, 7-1] and time-domain indirect method. The main goal of frequency domain approache i to compute all critical purely imaginary root of the characteritic equation and time delay value at which the ytem will be marginally table. The indirect time-domain method that utilize Lyapunov tability theory and linear matrix inequalitie (LMI) technique have been ued to etimate delay margin of the LFC ytem[1,, 11]. All exiting method aim to compute delay margin jut conidering tability a the only deign conideration and to etimate delay margin value at which the LFC ytem will be marginally table for a given et of PI controller parameter. However, a practical LFC ytem cannot operate near uch point becaue of unacceptable ocillation in the frequency repone. More importantly, a mall increae in the time delay could tabilize the LFC ytem if the time delay i around the delay margin. Therefore, in addition to the tability conideration, other deign pecification uch a gain and phae margin that enure a deired dynamic performance (i.e., damping, teady-tate error, ettling time, etc.) mut be taken into account in delay margin computation. A imple method that include a gain-phae margin teter (T) in the feedforward part of the control ytem ha been preented in [1] to analyze the gain-phae margin of time-delayed control ytem with adjutable parameter. Thi paper extend our earlier work reported in [3] to compute delay margin baed on pecified gain and phae margin. For that a purpoe, LFC model i modified to include a frequency independent T a a virtual compenator in the feedforward part of each control to add deired gain and phae margin to the LFC ytem with delay. Uing the exact method [3, 8], firt the delay margin value at which the modified LFC ytem with a T will be marginally table are computed for a wide range of PI controller gain. Then, uing thee delay margin, root croing frequencie of the imaginary axi and phae margin, time delay value at which the original LFC ytem without a T will have a deired gain-phae margin are analytically determined. Time-domain imulation reult indicate that gain and phae margin a an indication of tability margin hould be included in the delay-dependent tability analyi of LFC ytem with delay to achieve an improved frequency repone in term of le overhoot, ocillation and ettling time.. Modified LFC model with T The block diagram of a two-area LFC ytem with a communication delay into the control loop i hown in Fig. 1. For the tability analyi, the characteritic equation of two-area LFC ytem with time delay i required [3]. (, ) () () τ τ Δ τ = P + Q e + R() e = τ i the total time delay. P(), Q (), R() are polynomial in with real coefficient and depend on ytem parameter. Thoe coefficient are not preented due to inufficient pace.

2 τ jφ τ jφ Δ (, τ ) = P() + Q() e Ae + R() e A e = τ τ = P ( ) + Q ( ) e + R ( ) e = (4) Fig. 1. Fig.. Block diagram of the two- area LFC ytem with communication delay Modified block diagram of a ingle area LFC ytem with a T P () = p 9 + p 8 + p 7 + p p5 + p4 + p3 + p Q() = q6 + q5 + q4 + q3 + q + q1 3 R () = r 3 + r + r 1 + r In Fig. 1, Δf, ΔPm, ΔPv, Δ Pd are the deviation in the frequency, the generator mechanical output, the valve poition, and the load of the each control area, repectively. M, DT, g, T ch and R denote the generator inertia contant, damping coefficient, time contant of the governor and turbine, and peed drop of the each control area, repectively. ACE and ACE repreent the area control error and it integral. β i the frequency bia factor. Finally, T 1 denote the tie-line ynchronizing coefficient between the control area. The uer defined Gain and Phae Margin Teter (T) a a virtual compenator i added to the feedforward path in each control loop of the LFC ytem a hown in Fig.. The frequency independent T ha following form: () C( A, φ ) = Ae jφ (3) A and φ repreent gain and phae margin, repectively. The characteritic equation of the modified LFC ytem with the T given in Fig. i then obtained a P () = P() = p9 + p8 + p7 + p p 5 + p 4 + p 3 + p Q () = A q + q + q + q + q + q ( ) ( 3 1 ) R () = A r + r + r+ r Note that in (4), we have an exponential term e τ rather than e τ a in. Thi i obtained by combining e τ jφ and e into a ingle exponential term for = which i the complex root of (4) on the imaginary axi. The relationhip between τ and τ i given a φ τ = τ + (5) It mut be emphaized that the time delay value τ at which the characteritic polynomial of the modified LFC ytem with T hown in Fig. ha root on the imaginary axi i the tability delay margin of the modified LFC ytem, not the original LFC ytem. Therefore, we firt need to compute the purely imaginary root of the modified LFC ytem =± and the correponding delay margin τ. Then, uing thee, the time delay value at which the original LFC ytem will have a deired, A and φ, could be eaily determined uing (5). 3. Gain and Phae Margin baed Delay Margin Computation The tability of the modified LFC ytem with T i evaluated by the poition of the root of the characteritic polynomial of the form in (4). For aymptotic tability, all the root mut remain in the left half of the complex plane for a given time delay. Note that the characteritic polynomial of (4) i a trancendental equation due to the term in the form of e τ. A a reult of thi trancendental feature, the characteritic polynomial ha infinitely many finite root and the computation of thee root become quite difficult. However, the main objective here i to compute delay value for which the characteritic polynomial of (4) ha root (if any) on the imaginary axi. Aume that Δ ( j, τ ) = ha a root on the imaginary axi at = for ome finite value of τ c. Since complex root alway exit a a conjugate pair, the equation Δ( j, τ c) = will alo have the ame root at = for the ame value of τ c. A a reult, the delay margin problem now become a problem of finding value of τ c uch that both Δ ( j, τ c) = and Δ( j, τ c) = have the ame root at = jω c. Thi reult could be expreed a [8]:

3 jωτ (, ) ( ) ( ) c c Δ τc = P + Q e + jωτ ( ) c c R e = jωτ ( j, ) ( ) ( ) c c Δ τc = P + Q e + jωτ R( jω ) c c c e = The exponential term in (6) hould be eliminated to obtain a new characteritic polynomial without trancendentality. Thi could be eaily achieved uing a two-tep recurive procedure a decribed below. Let u define new characteritic equation at =± jω c a [8] and Δ (, τc ) = P ( ) Δ(, τc ) jωτ R( j ) c c e Δ (, τc ) = (, τc ) [ P ( ) P ( ) R ( ) R ( ) ] jωτ [ P( jω ) Q( jω ) R( jω ) Q( jω )] e c c = Δ = + c c c c Δ (, τc ) = P ( ) Δ(, τc ) jωτ R( jω ) c c c e Δ (, τc ) = (, τc ) [ P ( ) P ( ) R ( ) R ( ) ] jωτ [ P( jω ) Q( jω ) R( jω ) Q( jω )] e c c Δ = + c c c c = It i clear from (7) and (8) that the root = of (6) i alo a root of the following new characteritic equation. Another way of aying, the purely imaginary root of (6) i preerved in (7) and (8). New characteritic equation in (7) and (8) could be rewritten in a much more compact form a jωτ (, ) ( ) ( ) c c Δ τc = P + Q e = (9) jωτ ( jω, ) ( ) ( ) c c Δ c τc = P + Q e = P ( ) = P ( ) P ( ) R ( ) R ( ) Q ( jω) = P ( ) Q ( ) R ( ) Q ( ) (6) (7) (8) It hould be noted that the new characteritic equation in (9) j contain only a ingle e ωτ c c jωτ and e c c, indicating that the degree of commenuracy i reduced from to 1. On the other hand, the degree of polynomial P ( jω c ) and Q ( jω c ) now become 18 and 15, repectively after eliminating the term jωτ of e c c in (6). Thi procedure could be eaily repeated to jωτ eliminate exponential term, e c c j and e ωτ c c in (9) and the following augmented characteritic equation not containing any exponential term could be obtained. () ( ) () Δ = P ( ) = (11) () P ( ) = P ( ) P ( ) Q ( ) Q ( ) (1) It hould be noted here that the root (6) for τ c i alo a root of (11) ince the elimination procedure preerve the purely imaginary root of the original characteritic equation of (6). The equation in (11) can be eaily rewritten a the following polynomial in ω c. W ( ) = P ( ) P ( ) Q ( ) Q ( ) = (13) Note that the new characteritic equation in (13) without any exponential term ha the degree of 36 a given below. By ubtituting P' ( = j ),Q'( = ) and R' ( = jω c ) polynomial into (9)-(14) ubequently, we can obtain the augmented characteritic equation a follow W( ) = t36 + t t + t = (14) Due to inufficient pace, the coefficient of (14) are not preented. It i obviou from (14) that the exponential term in the characteritic equation given in (4) or (6) are now eliminated without uing any approximation. For that reaon, the poitive real root of (14), ω c >, coincide with the imaginary root of (4), =± exactly. For a poitive real root ω c, the correponding value of τ c of the modified LFC ytem with the T can be eaily computed uing (9) a follow [3, 8] : P ( jω ) c Im 1 1 Q ( j ) τc = Tan P ( ) Re Q ( ) Table 1. Delay margin reult for A = 1 and φ = K P Table. Delay margin reult for A = and φ = K P (15)

4 Table 3. Delay margin reult for A = 1 and φ = K P Table 4. Delay margin reult for A = and φ = K P Δf (pu) 6 x τ=3.631 τ=.799 τ= Time () Fig. 3. Damping effect of the gain and phae margin on the frequency repone for ( A = 1, φ = ), ( A = 1, φ = ), ( A =, φ = ) for K =., K =.4 P I Once we have found the tability delay margin of the modified LFC ytem with the T, uing (5), we could then eaily compute the time delay value for which the original LFC ytem will have the deired gain and phae margin a follow: τ φ = τc (16) φ i the deired phae margin and ω c repreent the root croing frequency. 4. Reult The ection preent gain and phae margin baed delay margin reult and verification tudie uing time-domain imulation. The parameter of the two-area LFC ytem are given in [, 3]. Delay margin that enure the deired gain and phae margin are computed uing (15) and (16) for a large et of PI controller gain and for variou gain and phae margin. Table 1 preent delay margin for A = 1 and φ =. Thi cae correpond to the conventional tability delay margin computation which the gain and phae margin are not taken into account. Therefore, τ = τ ince τ decreae a φ = a een from (16). Reult how that increae for a fixed K P. Thi indicate that the increae of caue a le table LFC ytem. Next, the gain and phae margin are elected a ( A =, φ = ) to invetigate the impact of the gain margin only on the delay margin. The correponding delay margin reult are preented in Table. It i clear that the incluion of the gain margin notably reduce the delay margin for all PI controller gain. The effect on delay margin of phae margin φ i alo tudied. Table 3 give the delay margin reult for ( A = 1, φ = ). Fig. 4. Combined damping effect of the gain and phae margin on the frequency repone for ( A =, φ = ), ( A = 1, φ = ), ( A =, φ = ) Similar to the gain margin cae, the reult clearly indicate that delay margin decreae for all PI controller gain when the phae margin i only conidered. However, the decreae in delay margin i le than that of the gain margin cae given Table. Finally, both gain and phae margin are included in the delay margin computation. Table 4 illutrate delay margin reult for ( A =, φ = ). When compared with the gain margin cae ( A =, φ = ) in Table and the phae margin cae ( A = 1, φ = ) in Table 3, it i obviou that the combined effect of the gain and phae margin on delay margin are much remarkable than their individual effect. Table 1 to 4 clearly indicate that for all PI controller gain delay margin decreae when gain and phae margin are taken into account. For a elected PI controller gain, KP =., KI =.4 for example, the implication of thee delay margin could be explained a follow: A can be een from Table 1, the LFC ytem will be marginally table at τ = 3.63, which mean the ytem will not have any tability margin in term of the time delay. A mall increae in the time delay will detabilize the LFC ytem. Table indicate that the LFC ytem will have the gain and phae margin of ( A =, φ = ) at τ = 1.8. That mean the ytem i not only table but alo ha the deired gain and/or phae margin.

5 In thi cae, the ytem will remain table if even a mall increae i oberved in the time delay. Time-domain imulation are performed to illutrate the reaon why the gain and/or phae margin hould be conidered in delay margin computation. A load diturbance of Δ Pd =.1 pu at t = 1 i conidered. Fig. 3 compare the frequency repone of the LFC ytem for ( A = 1, φ = ), ( A = 1, φ = ), ( A =, φ = ) and ( KP =., KI =.4). From Table 1, note that delay margin i found to be τ = 3.63 for ( A = 1, φ = ). A een in Fig. 3, the LFC ytem exhibit utained ocillation for thi delay value, which indicate that LFC ytem i marginally table. However, uch ocillation in the frequency deviation are not acceptable from a practical operating point of view. To eliminate uch an undeirable ocillation, we need to conider gain and/or phae margin in delay margin computation. A can be een in Table and 3, delay margin baed on gain and phae margin are computed a τ = 1.8 for ( A =, φ = ) and τ =.79 for ( A = 1, φ = ). A compared to the cae of ( A = 1, φ = ), it i clear from Fig. 3 that ocillation quickly damp out for τ = 1.8 when the gain margin ( A =, φ = ) i conidered. Similar obervation could be made for the phae margin cae, ( A = 1, φ = ) in which ocillation alo damp out but in a longer time. Finally, Fig. 4 illutrate the combing effect of gain and phae margin on the damping of LFC frequency repone for ( A =, φ = ). From Table 4, the correponding time delay value i computed a τ =.94. Fig. 4 clearly how that ocillation quickly damp out when both gain and phae margin are conidered. Thi imulation reult clearly indicate that gain and/or phae margin mut be included in delay margin computation to have an improved dynamic repone (fat damping, le overhoot, le ettling time, etc.) of the LFC ytem with time delay. 6. Concluion Thi paper ha utilized an analytical method to compute delay margin of a time-delayed two-area LFC control ytem conidering not only tability but alo gain and phae margin. It ha been hown that delay margin obtained conidering only the tability reult in unacceptable ocillation in the frequency deviation and delay margin baed on gain and phae margin improve the dynamic repone. Depending on the value of time delay oberved in the LFC ytem, the PI controller parameter could be properly elected uch that the two-area LFC ytem will be not only table and but alo will have a deired dynamic performance in term of damping, ettling time and nonocillatory behavior. 7. Reference [1] X. Yu and K. Tomovic, Application of linear matrix inequalitie for load frequency control with communication delay, IEEE Tran. on Power Syt., vol. 19, no. 3, pp , Aug. 4. [] L. Jiang, W. Yao, Q. H. Wu, J. Y. Wen, and S. J. Cheng, Delay-dependent tability for load frequency control with contant and time-varying delay, IEEE Tran. Power Syt., vol. 7, no., pp , May 1. [3] Ş. Sönmez, S. Ayaun, C.O. Nwankpa, An exact method for computing delay margin for tability of load frequency control ytem with contant communication delay, IEEE Tran. Power Syt., vol. 31, no. 1, pp , Jan. 16. [4] B. Naduvathuparambil, M. C. Valenti, and A. Feliachi, Communication delay in wide area meaurement ytem, in Proc. the 34th Southeatern Sympoium on Sytem Theory (SSST), Huntville, Alabama, March 18-19,, pp [5] P. Kundur, Power Sytem Stability and Control, New York: McGraw-Hill, [6] M. Liu, L. Yang, D. Gan, D. Wang, F. Gao, and Y. Chen, The tability of AGC ytem with commenurate delay, Euro. Tran. Electr. Power, vol. 17, no. 6, pp , 7. [7] J. Chen, G. Gu, and C.N. Nett, A new method for computing delay margin for tability of linear delay ytem, Sytem and Control Letter, vol. 6, no., pp , [8] K.E. Walton and J.E. Marhall, Direct Method for TDS Stability Analyi, IEE Proceeding Part D, vol. 134, no., pp.11 17, March [9] Z. V. Rekaiu, A tability tet for ytem with delay, in Proc. Joint Automatic Control Conference, San Francico, California, Aug , 198. [1] N. Olgac and R. Sipahi, An exact method for the tability analyi of time-delayed linear time-invariant (LTI) ytem, IEEE Tran. Automatic Control, vol. 47 no. 5, pp , May. [11] C. K. Zhang, L. Jiang, Q. H. Wu, Y. He, and M. Wu, Further reult on delay-dependent tability of multi-area load frequency control, IEEE Tran. Power Syt., vol. 8, no. 4, pp , Nov. 13. [1] C.H. Chang and K.W. Han, Gain margin and phae margin for control ytem with adjutable parameter, Journal of Guidance Control and Dynamic, vol. 13, no. 3, pp , 199.