IEEE Transactions on Power Apparatus and Systems, Vol. PAS-104, No. 12, December 1985 3409 DYNAMIC BEHAVIOUR OF AGC SYSTEMS INCLUDING THE EFFECTS OF NONLINEARITIES A. Norched, Member IEEE -Ontario Hydro Toronto, Canada 1.B. Shahrodi, Member IEEE University of Toronto Toronto, Canada Abstract - A general mathematical model capable of representing multi-area, multi-unit systems has been used to study the dynamic behaviour of AGC systems. The eigenvalues of the system are obtained and classified. The modes which play the most Important role in the system dynamics are identified and related to the system control loops. The effects of the nonlinearities are assessed by their position in the control loops. It is shown that the adverse effects of the nonlinearities are pronounced when the linear model at operating point, has lightly damped critical modes. The effects of the nonlinearities can be minimized by increasing the damping of these modes using methods outlined in the paper. 2. SYSTEM EIGENVALUES A functional block diagram of a one unit control area in an interconnected power pool is shown in Fig. 1 [1]. In the multiple loop control system of Fig. 1, where control actions take place at different speeds, dominant modes can be related to separate control loops (7]. The parameter changes within a particular loop, affect mainly the corresponding dominant mode. 1. INTRODUCTION Linear models have been widely used to study AGC The studies done were either systems [1,2,3]. low limited to order or systems were too to mathematically oriented offer a physical understanding of the system dynamic behaviour. As for the effects of the system nonlinearities, the number of published studies is relatively small. The lack of a general method for the study of large scale nonlinear systems has prompted a series of works using simulation techniques (4,5]. For this study, detailed multi-area, multi-unit AGC model, with the option of including the system nonlinearities, has been developed [61. A large number of AGC systems, with various configurations and control are strategies, examined. The eigenvalues of the system are categorized and related to the structure of the AGC model. The oscillatory dominant modes are identified. These modes are referred to as 'critical modes'. Methods to improve the stability of the system through increasing the damping of the critical modes are presented. Simulation results are used to compare the dynamic behaviour of the nonlinear system with that of the corresponding linear model. The time response of the system is obtained for step load changes representing emergency conditions as well as for morning pick-up conditions. The latter is simulated by low-amplitude random signal superimposed on ramp loads. The conditions under which the nonlinear effects become dominant are identified and methods to alleviate these effects are presented. Figure 1: Control loops in an interconnected area. a the of phenomena results The study explain encountered in AGC systems such as the commonly observed tie-line oscillations. 85 WM 073-2 A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the IEEE/PES 1985 Winter Meeting, New York, New York, February 3 8, 1985. Manuscript submitted August 10, 1984; made available for printing November 28, 1984. - The natural modes of an AGC system can be classified as: (a) (b) (c) (d) a. Critical Modes, Slow Non-Oscillatory Modes, Fast Non-Oscillatory Modes, and Well-Damped Oscillatory Modes. Critical Modes The critical modes are lightly damped oscillatory modes which are subject to instability as the AGC controller gains are varied. They are pronounced in most of the system variables and play a significant role in their time responses. These modes are: (i) Af-mode, (ii) APtie-mode, and (iii) AGC-APtie-mode. Some of the frequently observed tie-line oscillations can be attributed to the poor damping of one or more of these modes. Af-mode Af-mode is an oscillatory mode mainly associated with the governor loop. The frequency of this mode, is in the range of 0.01-0.3 Hz. This wide range of frequency is due to the difference in the type of units. The faster the response of the unit, the higher the frequency of the Af-mode and the more damped this mode is. 0018-9510/85/1200-3409$01.00©1985 IEEE 3410 AP tie-mode This oscillatory mode is mainly associated with the 'natural loop'. The frequency of this mode is the highest among the frequencies of the critical modes and is in the range of 0.2-0.6 Hz. APtie-mode is typically poorly damped. This mode represents the The electromechanical oscillations of the areas. faster the response of the units, the more damped the APtie-mode is. Table 1 AGC System Parameters fo Nominal frequency Turbine time constant Governor time constant ACE Smoother time constant Governor droop Transient regulation Proportion of power developed before reheater Reheater time constant Nominal load in an area Synchronizing coefficient in the MW base of area 2 Water starting time Transient droop time constant Speedchanger servomotor time constant for a non-reheat unit for the first reheat unit for the second reheat unit for a hydraulic unit Frequency -bias gain Proportional gain Integral gain Unit inertia constant for a non-reheat unit for the first reheat unit for the second reheat unit for a hydraulic unit TT TG TSM R r C TR AGC-APti e-mode PB T12 The frequency of this oscillatory mode is the lowest When there is no among the critical modes. integrator in the AGC controller, this mode loses its The AGC-APtie-mode is almost oscillatory form. independent of the governor and AGC-Af-loops. Unlike the other two critical modes, this mode is not subject to instability as the AGC integrator gain The frequency of this mode is in the increases. The damping and the range of 0.001-0.02 Hz. frequency of this mode are highly dependent on the AGC controller gains and the speed of unit response. TW Tr b. Slow Non-Oscillatory Modes The presence of these modes is the result of the slow-response components of the system such as a hydraulic unit with a transient droop governor. The hydraulic slow mode is dominant in the time response It is almost of the hydraulic unit output. insensitive to the variations of system parameters. c. B kp kj H Table 2: 8.0 sec. 0.5 pu. 0.1 pu. 2.0 8.0 sec. sec. 0.5 sec. 0.8 sec. 0.6 sec. 0.3 sec. 0.3 0.13 0.13 5.0 8.0 6.0 3.0 sec. sec. sec. sec. Eigenvalues of a Two-Area System With and Without Linearized Boiler Models Fast Non-Oscillatory Modes These modes appear due to the fast-responding components of the system such as the turbine, the permanent droop governors, and the servomotors. The fast non-oscillatory modes are almost insensitive to the variations of the system loop gains. Some of them are almost equal to the open loop poles of the fast-responding components. d. TSR 60.0 Hz 0.3 sec. 0.08 sec. 0.3 sec. 3.0 Hz/pu. 24.0 Hz/pu. 0.3 Well-Damped Oscillatory Modes Similar to fast non-oscillatory modes, these modes do not play a significant role in the response of the system due to their high damping. In this group of modes, only the AGC-Af-mode is worth mentioning. This mode is associated with AGC-Af loop and appears in oscillatory form only when the total area inertia is low (less than 10.0 s in. this study). The frequency of this mode is in the order of 0.02 Hz and its damping ratio is in the range of 0.6-0.9. This mode is pronounced in the response of the area frequency and the unit output, especially when the Af-mode is highly damped, i.e., when the unit is fast-responding. SOURCE OF EIGEN VALUES OR MODES NO BOILER EFFECT INCLUDED LINEARIZED BOILER MODEL hf-mode -0.0652±jO .1685 -0.1139±jl1.1838 .079±j0.0775 -0.0648+j0.1653 -0.0190,-0.0236 -0.0189,-0 .0217 -0.125,-0.1586 -0.2993,-0.3883 -0.9374,-0.9643 -1.250,-1.249, -1.667,-2.005, -3.333,-3.333, -3.333,-1.999 -O.125,-0.1586 -0.3043,-0.3874 -0.9380,-0.9644 -1.250,-1249, -1.667,-2.005, -3.333,-3.333, -3.333,-1.999 -3.333,-3.333, -3.333,-3.092 APtie-mode AGC-APtie-mode Hydraulic slow mode Reheater mode ACE smoother Hydraulic turbine Servomotor Steam turbine Governor Boiler Sample Results The eigenvalues of a two area system are calculated. Area 1 consists of one non-reheat unit, two reheat units and one hydraulic unit and Area 2 consists of one non-reheat unit, one reheat unit and one hydraulic unit. The system parameters used are given in Table 1 and the calculated eigenvalues are classified in Table 2. The eigenvalues shown are for the two cases: a) when boiler dynamics are ignored, and b) when the linearized boiler model given in the appendix is included. Figure 2 shows the time response of the incremental frequency of each area to a 0.1 pu. load change in -O -3. 333,-3.333, -3.333,-3.084, -2.998 -12.5,-12.5 -12.5,-12.5, -12.5,-12.596, -12.652 -0.1104+j 1.1818 -0.0:777±jO.0718 -3.007 -12.5,-12.5, -12.5,-12.5, -12.5,-12.594 -12.694 -O.0078±jO.0110, -O.0079±jO.0109, -O.0118±jO.0107, -O.0118±jO.0113, -O.OlOl+jO.0112, -O.0973±jO.0623, -0.0869+jO.0788, -0.0900+jO.0818, -0.0912+JO.0718, -0.0964+j 0.0630, -O.3012,-0.2875, -.2954. -0.2954±jO.0003 The high-frequency oscillations of the and the low frequency oscillations of Af-mode can clearly be seen in these figures. area two. APtie-mode _ tS_. j, 90% :: : . r FREOUENCY OF AREA * 1 Figure 3: ..._ . ... .... ~~~~~~~~~~~~. 2't ' -_,,. 1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.... ... a . 7i. .. li.s0 0 9I.W 2 Figure 2: Time response of 0.1 pu, step load 3. a two-area change in system area -Valvre po3itin_ Variation of the coeCficient 'k4' as a function of unit loading. of the position feedback of the causes servomotors instability of to In order AGC-Af-modes. and AGC-APtie restabilize the system, the following measures can be taken: The opening speedchanger to 2. fast-responding units are dominant in the area, the integrator should be removed from the AGC controller. In case it is desired to damp oscillations electromechanical the out (APtie-mode) as well, the unit MW-feedback (a) If DYNAMIC STABILITY OF THE AGC SYSTEM The results of the study show that the dynamic stability of the system is governed by the amount of damping of the system's critical modes. The effects of structural and parameteric changes on the damping of these modes are sumnarized below: I. - feedback. TlIM IN SECONDS FREQUENCY OF AREA :: t80~-v --- |: -) Throughout the study, it is assumed that all the speedchanger servomotors are equipped with position .... .-r- -4----*----... ----_-_,,,, - 4- b .*..f--t Minor Feedback Loops III. 1!{ : : :::: 3411 Loading of the Units Higher loading of the units results in more damping of the critical modes. In other words, a system with fewer units loaded to near their maximum continuous rating is more stable than a system with a larger number of lightly loaded units. II. Boiler Dynamics The nonlinear model of a drum-type boiler along with its linearized form are described in the Appendix. The linearized model of each boiler introduces five of Most the boiler additional eigenvalues. For the sample eigenvalues are lightly damped. system of section 2, the system eigenvalues with the inclusion of the boiler model are given in Table 2. It is clear from the table that the presence of the boiler has a negligible effect on the system's (dotted line in Fig. 1) is implemented. (b) If the integral controller remains in the AGC loop, the unit MW plus frequency feedback should be implemented. IV. The Addition of a New Generating Unit The addition of a slow responding generating unit to the AGC system results in the reduction of the damping of the critical modes. However, the addition of a fast-responding unit has a negligible effect on these modes. V. Interconnection of Areas The interconnection of two areas with lightly damped Af-modes results in a poorly damped or unstable other In words, slow-responding APtie-mode. units should not be dominant in interconnected a of interconnection the areas. However, fast-responding area to a slow-responding area may result in a stable APtie-mode. eigenvalues. Interconnected areas of equal sizes have a poorer dynamic performance than those with different sizes. An increase in the size of one of the interconnected The simulation results indicate that the boiler modes APtie-modes to increase. only in the time response of the variables inside the boiler control loops. The boiler effect the variables outside the boiler model is on negligible and can be accounted for by considering (see only the gain K4 relating Ams to AV Appendix). This gain accounts for the di?ference between drum and throttle pressures. Figure 3 shows the variation of k4 as a function of the valve opening (or the unit loading). Only when the boiler is loaded close to its nominal output is its effect pronounced. It is noted that the examined boilers are of drum-type with medium size drums. appear areas causes the damping of the Af- and VI. The Number of Units on AGC The opening of the AGC loops for some of the system units increases the damping of the Af-mode and the This is especially APtie-mode significantly. true when the units without AGC are the larger ones. Therefore, it is advantageous to use smaller units for AGC leaving the larger ones without AGC. This is consistent with the industry practice where very large units are not normally subjected to continuous and high-frequency control signals. 3412 Three-Area System Tie-Line Topology VII. The topology of tie-line connectilons is dictated by factors other than AGC. However, it is worthwhile to examine it.s effects on the AGC system stability. Figure 4 shows three power pools each consisting of three control areas interconnected by two tie-lines The connection of as shown by the solid lines. areas 2 and 3 (by the dotted line) has little effect on the Af-mode due to the low sensitivity of this The cases (b) mode to the APtie control action. and (c) of Fig. 4 are more prone to stability problems as they include the hydraulic unit with The transient droop which has a slow response. connection of the areas 2 and 3 causes stability problems in these two cases. 1 NR 1 RH,RH 1 NR 2 Effects of the Boiler As shown in Se ction 3, the linear drum-type boiler model does not play a major role 'in the dynamilcs of the thermal unit. The boiler control loops act almost indlependently of the AGC control, loops and have their own dynamics and eigenvalues. The nonlinear boiler m'odel is found to have si'milarly negligible effects on the AGC system [6]. Figure 4: Three-area pools examined. NRH = Non-reheat Unit RH = Reheat Unit HYD = Hydraul'ic Unit 4. Limit cycles result in inicreasing the RNS vallue~of the control error as well as the we'ar and tear in the system hardware. The abrupt changes in the unit output can be sufficientl'y damped, if the line'ar part of the system, follow ing the nonlinear element, acts as a low-pass filter. Consequently, abrupt changes due to the governor backlash a're more attenuated in the of units than output slow-responding fast-responding ones. On the other hand, the low frequency oscillations are not attenuaLted, silnce the linear part of the system, as a low-pass filter, cannot affect such silgnals. A large number of AGC systems with various system structures aknd operating conditilons are examined. The conclusions reached are listed below. I. 33 3NR (H) (A) (2) undesirable system response like abrupt changes in unit'outputs, and (3) low frequency oscill'ations leading to a sl'ow-down in the operation of the control loops. Figure 6 shows the response of an isolated non-reheat unit to a step of 0.4 pu. load cha'nge. Despite this abnormally large load change, the nonlilnear boiler model has a neglilgible effect on the dynamics of the THE EFFECTS OF NONLINEARITIES The dynamic behaviour of the AGC system under the effect of nonlinearities is related to the dynamics of its model linearized around the operating point. Approxim'ating a nonlinear element by a variable gailn, ki,: a necessary but not sufficient condition for the syste'm instability is that ki would produce instability in the equivalent linear system [8]. in sections 2 and 3, the dynamics of the AGC model was shown to be dominated by the system critical modes. These modes may become un'stable as a result of control loop gain variations. A nonlin'ear element can cause instability, if 'it provides a condition under which one of the critical modes becomes The condition for an unstable critical unstable. mode depends on the locatilon of the nonlinear elemen-t in the system and the course of possible equivalent gain (ki) variation. Filgure 5 shows the locations of some of the AGC nonlinear elements (9,10]. state variables outside the boiler model. However, its effect is pronounced in the state 'variables associated with the boiler (dr'um pressure in the figure). The figure also shows that the linearized ............ ..l .. 0.0 10.10 30.00 10.00 FREOUENCY OF AREA 0.10 5080 TIME IN SECONDS U.0so 7.080 U.W 00.00 70.00 lIMO00.0 00.00 1 zU~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- '0.. ..... 08.10 08.0 Figure 5: In addition nonlinear form (1) AGC system nonlinearities. to stability the element can problem generate limit DRUM PRESSURE DEVIATION OF stated above, cycles in a the of: steady state undamped conditions, osc-illations 30.00 around the steady 0.00 50.o TIME IN SECONDS UNITo1 Figure 6: The response of an isolated reheat unit to 0.4 pu. step load change. (x) nonlinear boiler model. (*) linearized boiler model. (+) coefficient k4 only. (A) without any boiler effects. 3413 thea and model approximate model (with only coefficient k4 considered) have virtually no effect the system dyn'amics, verifying the results on obtained in section 3. a k. II. Effects-of the Control Signal Deadband I A''A 'i ... TINE IN SECIONDS OF FREQUENCY AREA ..1. ' 1~~ ci .. .. ILU ~~~~.... .. .... .. (b) with E.;. w !W. .f\ 1 --- ci ua A -NJ V;v V. V, V" =a .........................n.....................V ...v .....VV.\I.. ...v cc,;. 9- I.M HAD to.60 FREQUENCY Figure -j OF 8: Figure The 7: of response morning pick-up The ~Study in deadband of width 0.02 deadband of width 0.008 of the Governor the of 'and APtie-modes AGC linear Af- the gain model lightly are ratio of backlash may 8a,_ changes pic'k-up governor the the of of limit the output shows where the of width 0.008% pu. pu. backlash. no 03 W 0. 33 3.3.. . 0.0 3.00 .... or may linear figure. the parametric or leading condition. backlAsh .0 .. [N... NON GENERATION OF ....T.UN ....EC .T . W. S... .N....... Figure ...10 .The response........ of.n.soatd.aeaofon one reheat.. unit..... to... a an non ehea increases. the of one on governor (Fig. cycles to response the the power abrupt backlash 9) generating changes, interesting output and/or of the for effect units low-amplitude load changes can clearly be seen in output changes in a stepw'ise manner. This an .....T.I.... of -leading stable The to system MW-loop APtie-mode ....... An of the .W. The backlash effects model unit of two-area a change. this For The response of an isolated area of one reheat and one hydraulic units to a 0.1 pu step load.change. (x) w-ith backlash. (6) no backlash. the of by 1 FREQUENCY OF AREA Figure 9: Therefore, the the system. such high-frequency traceked. of backlash 8b. effect in in seen the of 'in Fig. Figure 10. backlash with a lowest the under under damping. of restabilization Another has modes. load under implementation the .appearance step the of its damping shown with 1.. the Atf- backlash response stabilized be in improvement be mode changes structural the pu. clearly can critical is loop. when if the instability to APtie-mode of unstable the the to reduced critical shows 8a 0.1 a instabil'ity Fig. high a governor Thus, i-mode Figure to effect showed the damped, the among ~~~~~~~~~~~ ~~~~~~~~~~~~~prone system (x) to 2. pu. APtie-modes and variations two-area effect. system two-area change load pu. unstable. them a a 'in area wildth 0.02% step Backlash these' modes are of damping decreases. loop gain In of response pu. a 1.. governor damping - TliN [N SECODS sensitivity make to deadband. no of parameter .The 70.80 "-m 2. area +)with Effects The system two-area a condition (6) 'With (x) III. .. IN SECONDS 0.1 (&) . .. AREA OF FREQUENCY IN SECONDS TINE The Wu. YU GU. uu 19. w 10.00 10.3. AREA TINE .... .. the MW-feedback. U-M M-M Wan As nn 21 0V (a).......... ..W W .W 3 .W 2 0.3 11 x A 3~~~~~~~~~~~~~~~~~~~. CCj - t ...I h.PI1/'.~~~~~~V Vv v v v rt The control silgn'al deadband approximated as a variable gain, reduces the, AGC loop gain. Therefore, the stability of the system is not jeopardized by the The critical modes presence of this deadband: to namely instability, subject APtie- and hf-modes, do not bee-ome unstable as a result of AGC control loop gain reduction. As for the response of the system to morning pick-up conditions, the results show that the system load tracking deteriorates and the RMS of the control error incr'eases apprecilably (Fig. 7) under the effect of the deadband. I ...:....... effect is the abrupt plants. mornilng the of is that are Fig. not 10 rat liitsoaeareahighcopaed or gThe rseroneo tothe rate ~~~~ofceheang oflod unernormaeopeatnt ting oremergency pic-upcondiios. Thre oren herefeto Thegvale 3414 *, the system response can be ignored. This is verified for step load changes (Fig. 11) as well as morning pick-up conditions. . N tL IL JE -- *- ;-'''*,'*-'';-''*''--''';..'..''..'..'..''.. '... *. '-.. ;'.'*... -. ",.. -..'*.....'*.; 6 U._: : . 1@LM HADZ Figure 11: U.0 w. i__ .... ..... N' ......O _ . ION Of on 5. Ks-_-__,. ..:.. ** REHEAT ; ........ UNI T C~~~~~~~~ENERAT ................ ............... o ..... an aTIN U T iME IN SECOMD iani. n N U.E "N As an The response of an isolated area of one reheat and one hydraulic units to a 0.1 step load change. (x) with valve rate limits. (+) with speedchanger rate limits. (A) no nonlinearity. the other hand, the speedchanger rate limits play major role in the unit response, especially for thermal units where the limits correspond to +1-4% of their maximum continuous rating (MCR) per minute. For hydraulic units the limits can reach +100% of the unit MCR per minute. The effects of speed limiters are summarized below: On a (1) The speedchanger rate limiter has almost no effect during the normal operating conditions. (2) The stability of the system is not threatened by this limiter since the Af- and APtie-modes do not become unstable when the AGC loop gain is reduced. (3) The speedchanger rate limiter causes an appreciable slow-down in the AGC control action. This effect is pronounced when the rate limits are low or the rate of change of the load is high (Fig. 11). Low frequency limit cycles, referred to here as AGC-limiter oscillations emerge in such cases. V. The Simultaneous Effects of the Speedchanger Rate Limiter and the Governor Backlash __. . -, .................-A .......O. M * 14 64 P ~ i -. .. -. .._ _ t- . . F :. FI1- .: ..........UNIT , ......N ....GENERATION OF NON-RHT UNIT IN ....... AREAwl ', ............. TIME IN SECONDS Figure 12 The response of an isolated area of one non-reheat and one hydraulic units to a 0.1 pu step load change (A) with speedchanger rate limits and the backlash. (+) with speedchanger rate limits only. (x) no Simulation studies were carried out for step load changes representing emergency conditions and ramp loads with random components representing morning pick-up conditions in a wide range of system configurations and operating conditions. The results show that: the presence of nonlinearities can result in destabilizing the lightly damped critical modes of the system. The system regains stability when the damping of these modes is increased using suggested measures. Nonlinearities also result in undesirable abrupt changes in unit outputs and/or cause an increase in control errors. ACKNOWLEDGEMENTS The authors wish to express their thanks to Prof. A. Semlyen of the Department of Electrical Engineering, University of Toronto for his support throughout this study. Thanks are also due to Mr. P.L. Dandeno of Ontario Hydro for his suggestions during the preparation of this paper and to the NRC for financing the computer expenses. REFERENCES [1] 121 [4] [5] ---- -4 nonlinearity. CONCLUSIONS An AGC model for multi-area, multi-unit systems was developed to study system dynamics with and without the effects of nonlinearities. Based on the eigenvalue analysis, critical modes were identified and related to the system control loops. The effects of the system configuration and the loop parameters on the damping of these modes were examined. [3] The step responses of the examined systems show that the simultaneous effects of the rate limiter and governor backlash is mostly a combination of their individual effects. Generally, when the rate of change of the load is high, AGC limiter oscillations dominate. Figure 12 shows the high-frequency backlash limit cycles superimposed on the low frequency AGC limiter oscillations in response to a 4*. step load change. On the other hand, the backlash limit cycles are pronounced when the rate of change of the load is low. Thus, for a simulated morning pick-up, the system response is only under the backlash effect. O.I. Elgerd, Electric Energy Systems Theory, McGraw-Hill Book Co., 1971, Chapter 9. G. Quazza, control "Non-interacting of interconnected power systems", IEEE Trans. Power App. Syst., vol. PAS-85, pp. 727-741. G. Quazza, "Criteria for equitable participation of areas in tie-line power and frequency control of an interconnected power systems", Automazione e Strumentazione, March 1966. F.P. deMello, R.J. Mills and W.F. B'Rolls, "Automatic generation control, part I: process modelling", IEEE Trans. Power App. Syst., vol. PAS-92, pp. 710-715, March-April 1973. R.A. Schlueter, J. Perminger, et al., "A nonlinear model dynamic of the electric coordinated systems", IEEE Winter Meeting, Paper No. C75088-0, New York, N.Y., January 26-31, 1975. E.B. Shahrodi, "Modelling and analysis of AGC N.A.Sc. systems", Thesis, of Department Electrical Engineering, University of Toronto, January 1981. [7] John W. Brewer, Control Systems Design, Analysis and Simulation, Prentice Hall, Inc., 1974, Chapter 16. and [81 R.E. "Physical mathematical Kalman, mechanisms of instability in nonlinear automatic control Trans. Vol. systems", ASME, 79, pp. 553-563, 1957. [9] IEEE Committee, "Dynamic models for steam and hydro turbines in power systems studies", IEEE Trans., vol. PAS-92, pp. 1904-1915, Nov./Dec. 1973. (10] C. Concordia, L.K. Kirchmayer, E.A. Szymanski, "Effect of speed-governor deadband on tie-line power and frequency control performance", AIEE [6] 3415 Trans. on Power Apparatus and Systems, vol. 75, pp. 429-435, August 1957. APPENDIX and the quantities at the operating point are denoted by a subscript 'o' (PDo, PTo, etc). The values chosen for the storage constant, cb, are between 90 and 300 seconds [41. The definition of the boiler parameters and their typical values are given in Table Al. Boiler Model The drum-type boiler model used in the study is shown in Figure A.l. Table A.1 Data for Drum tyRe Fossil Fuelled Boiler Tw Tf TD Tr Waterwall time constant Fuel system time constant Fuel firing system delay time Lead-lag compensator time Boiler integrator gain Proportional-integral ratio of gains Superheater friction drop coefficient Throttle pressure setpoint Valve position setpoint Steam flow setpoint Boiler storage constant KI TI K PT0 PVO msO Cb 'igure A.1: The nonlinear boiler model a drum-type boiler, the throttle pressure PT is ferent from the drum pressure PD by the rheater drop PDT. This drop is given as su PT PD where mS = Vp = is = k (1) 2s (2) VPPT valve position, steam flow, and the superheater friction drop coefficient. = k PDT = = Equations (1) and (2) are included in the simulation using the boiler model shown in Fig. A.1. process In this figure, fl(PD,Vp) = f2(PDV) = +4k Vp Vp f1(P 1) /(2kVp VP) , The model of Fig. A.1 is linearized pT s where =l 2 as shown below: APD + k2 AVp k3 APD + k4 pV V2 bf 1 /bpD= (1+4k k1= 1 D k ) ~f /bVp 1 = k4 = bf2/bVp (1+4kc kI k3 =6f2JbPD = Po VPO 2 PDo -1/2 p -1/2 PDo) Po pTo+ k2 PO g 2~/V 20 2Apo2 /VVP 4.0 sec 10. 0 sec 6.0 sec. 69.0 sec. 0.03 26.0 0.06 1.0 pu. 0.5 pu. 0.5 pu. 90.0 300.0 sec.