PARAMETRIC ANALYSIS OF SHAFT TORQUE ESTIMATOR BASED ON OBSERVER

PARAMETRIC ANALYSIS OF SHAFT TORQUE ESTIMATOR BASED ON OBSERVER Tetsuro Kakinoki, Ryuichi Yokoyama Tokyo Metropolitan University t.kakinoki@h4.dion.ne.jp Goro Fujita Shibaura Institute of Technology Kaoru Koyanagi TEPCO Systems Corporation Toshihisa Funabashi Meidensha Corporation Abstract Under the action of disturbance in power system, turbine-generator shaft are subjected to mechanical stress caused by severe torsional torques. Shaft torque is proportional to the shaft twist, which is caused by angle difference between both ends. However, observing the angle difference in an actual generation system is difficult technically. On the other hand, measuring rotational speed of the shaft ends is relatively easy. Then if the shaft torque could be estimated effectively, it would be used in order to control the rotational speed for shaft torque reduction. This paper proposes a shaft torque excitation control for rotating machine by making use of an observer for the shaft torque. Firstly, the relationship between the observation signals and the accuracy of the estimation is discussed. Especially, the range of the error of actual values and measurement values of a turbine parameter are analyzed. Secondly, a shaft torque reduction control system is developed by using excitation control based on the estimate by identity observer. A co-generation 7-mass system is used for the test model, and Park s equation for synchronous machine is employed. Linear quadratic regulator theory (LQR) is employed to design the observer gain, which provides acceptable estimation. Keywords: torsional torque, torque estimator, observer, LQR, AVR, co-generation system, turbine parameters 1 INTRODUCTION Torsional oscillation in the rotating multi-mass shaft system has received wide attention. This problem has received much attention since the two incidents of generator shaft failure at Mohave Power Station, U.S.A. [1]. The destructive nature of these oscillations prompted many studies into understanding the factors affecting them as well as promising techniques to counteract them. The content of the research is classified as follows; 1) Mutual intervention with control system. 2) Sub-synchronous resonance with series capacitor compensation transmission system. 3) Shaft fatigue by system switching. For example, modal analyses in an actual operating condition at the accident location [2][3], and investigation of influence on shaft torque by interception condition [4] etc. are reported. The countermeasures for self-excited shaft torsional oscillations includes a wide variety of concepts and techniques. As countermeasures for shaft oscillation in thermal plants, static line filter, dynamic filter, and excitation control are used. For example, the resonance of the transmission system and the generating unit is avoided by the capacity control of the series capacitor for system compensation. The method of mitigating torsional torque by the excitation control has been reported as the countermeasures on the generating unit side. Since the excitation control is an optional function for automatic regulator (AVR) and power system stabilizer (PSS), it is regarded as the most effective method. Reference [5] introduces the design of the excitation control system by LQR observing all the state variables. However, the estimation method is not deeply except kalman filter, and the control system is provided by LQG [6]. The foregoing papers mainly describe thermal units. A similar problem can be researched also on a private generation plant system. A gas-turbine generator system in co-generation system has reduction gears and shear pins between the generator and turbine. The shear-pin works to prevent the generator from breaking caused by over load. However, once the pins break, the restoration takes long time. Therefore, some techniques to reduce the shaft torque and protect generator and shear pins are required. However, disconnecting of the generator from the power system is a typical method against system faults in practical power systems. From the above-mentioned background, the above technique can also be applied to gas-turbine generation system in reducing the shaft torque. However, measurement of the shaft torque inside the rotating machine is extremely difficult. Therefore, it is necessary to establish an estimation method of the internal torque. This paper proposes an observer based excitation control for rotating machine in power system, and a parametric analysis for shaft torque estimator based on observer against changes of turbine constants.

2 OBSERVER FOR EXCITATION CONTROL OF TURBINE-GENERATOR SHAFT OSCILLATION In this research, we aim at the establishment of torque reduction by the excitation control system which can be used from the private generation unit to the thermal generation system. Fig.1 shows a schematic diagram of the proposed excitation control system. The accurate measurement of the torque is necessary for the effective torsional torque reduction control. Normally, we cannot be measured, because it is covered by a case. Therfore, measuring rotational speed of the shaft ends is relatively easy. 2.2 Turbine Shaft Model Fig.3 shows the turbine-generator shaft model comprised of seven equivalent rotational masses, where joints, shear-pins, and an exciter are connected on the shaft. The shear pins are designed to be breakable against a severe damage, but in order to prevent the breakdown, the shaft torque should be minimized even in a severe damage. There are inertia constants of the rotating mass system, damping constant, and spring constant of the shaft as parameters, which decide dynamic response of the shaft torsional system [4]. Table I represents the mechanical constants of each mass. The inertia constants of turbine and generator are relatively large and dominant to characterize the shaft dynamics. The six shafts behave like a spring between masses. Therefore, a model is formed with seven masses connected with six springs. Figure 1: Observer System for Excitation Control of Rotating Machines. In this study, 3 different models; network model, turbine shaft model, and generator model are configured. 2.1 Network Model An in-house system as shown in Fig.2 which is designed by Meidensha Corporation in Japan is used for studies. Figure 2: Configuration of Private Generation System Model. The following is series of events applied as the test system disturbance[7]. step : steady state operation. step 1 : fault between feeder and transformer at.1 [s]. step 2 : disconnect the fault point. step 3 : disconnect general load. Figure 3: Turbine shaft model with seven equivalent rotational masses. Mass Inertia constant [s] Spring constant [pu/rad] 1 2.122 274.1982 2.811 1186.9 3.233 5.6128 4.44 264.279 5.44 28.3372 6.5688 21.677 7.87 Table 1: Mechanical constants of generator shaft. Equation (1) shows relationship of the mechanical input and output at the i-th masses, where, M is the inertia constant of each mass, D is the damping constant, K is the spring constant of turbine shaft, ω is rated speed, and denotes deviation from nominal value. Therefore, M i ω i is accelerating torque in i-th mass, D i ω i is damping torque and T i is input torque of each mass. Torque T i is positive if it is applied as an input mechanical torque in the turbine, and negative for the electric power torque in the generator. The torque propagates on each mass through the shaft connected on one or both sides. Then, the torsion corresponding to the spring constant is generated in the shaft, resulting in the rotational speed of each mass. d ω M T D ω K ( δ δ ) K ( δ δ ) (1) i d dt dt δi = i i i i1, i i1 i i, i 1 i i 1 = ω ω (2) i 2.3 Generator Model [5][8][9] The specification of the generator is shown Table II, where the parameters are typical values used for co-

generation system. The q-axis constants equal to the d- axis constants, i.e., X q =X d, X q =X d, and T qo =T do. X q and T qo are not considered. Item Value Rated frequency : f G 5 Hz Rated capacity : P G 5. MVA Rated : V G 6.6 kv Number of poles 4 Armature resistance : R a.64 pu Armature leakage reactance : X L.937 pu d-axis synchronous reactance : X d 2.27 pu d-axis transient reactance : X d.21 pu d-axis sub-transient reactance : X d.173 pu d-axis open-circuit transient time constant : T do 3.1 s d-axis open-circuit sub-transient time constant : T do.387 s Table 2: Generator parameter data. Generator's revolving speed Figure 4: Generator's terminal Figure 5: Reference speed Setting value of loads 16.666 1.3s Regulating rate 6 [%] Governor model 1. 1.5s Filter AVR model Reference 3.3 Setting value of 1. 1. 1. 1.326s 1. 1. 1. Mechanica Output Brushless EF PM Field voltag The transfer function of the governor is shown in Fig. 4. And, AVR used in the model is shown in Fig. 5, which is a normal model. 2.4 Reclosing Timing Fig. 6 shows the shaft torque generated between masses with a normal AVR (the supplementary excitation control is not used here). Simulation study is carried out with the system model mentioned previously. All lines except for generator-exciter are oscillating heavily due to the torsional torque of five shafts propagating on the turbine-generator unit masses, which are almost equal. A small oscillation exists on generator-exciter shaft. Exciter mass is not so mach affected by the torsion between turbine-generator units, since the exciter is relatively heavy generator mass. Therefore, the oscillation torque is also relatively small, but is not easy to stabilize. Step2 and step3 are at.15[s] and.33[s] respectively. These timing are adjusted for the torsional torques to become minimum value [1][11]. Oppositely, the torsional torques at the worst timing (.12[s] and.32[s] respectively) are shown in Fig. 7. In this case, each torsional torque are accumulated following the steps. The best timing is applied as a test scenario in the following simulations. Step1 Step2 Step3.6.4.2 -.2 -.4 -.6.1.2.3.4.5.6.7.8.9 1 Figure 6: Torsional shaft torques at the best timing..6.4.2 -.2 -.4 -.6 Figure 7: Step1 Step2 Step3.1.2.3.4.5.6.7.8.9 1 Torsional shaft torques at the worst timing. 3 OBSERVER DESIGN 3.1 Identity Observer Fig. 8 shows the full-order observer used here, which feedbacks output errors between the actual system x = Ax Bu (3) and the observer model x ˆ = Axˆ Bu (4) Here the closed-loop observer system is presented by xˆ = Axˆ Bu L( y Cxˆ) (5) where L is the observer gain matrix. The observation matrix specifies observable values among the estimated value xˆ. The observation matrix C specifies observable variables among the state variables. Three different cases, both side, turbine side, and exciter side of the shaft, are considered. The difference of A matrix in target system and A * in the observer is used to consider the error of measurement of the turbine shaft parameter. The error of the designed value and actual value under operating condition is reflected in each constant relatively [12]. In A*, the inertia constant and spring constant have small error [13] described following. As for damping effect,

damping constants are eliminated from A* because these varies depends on operating condition. u= Figure 8: Observation object B ẋ Identity Observer B. x^ 1_ s A(x,t) Observer gain 1_ s x^ C A*(x,t) x LQR L C Observation matrix Estimated value x^ The error of measurement is considered State estimation by identity observer 3.2 LQR Design The estimation error, e = x xˆ, satisfies the following equation: e = ( A LC )e (6) Linear Quadratic Regulator theory (LQR) is applied to design the observer gain L. In a controllable and linear time-invariant system, a state feedback control is achieved to minimize a performance index, t f T T J = [ x Qx u Ru]dt (7) where Q (n n) is a semi-positive definite symmetric weighting matrix, R (m m) is a positive definite symmetric weighting matrix. Under the performance index given by (7), an optimal control input for the terminal time t f is given by 1 T u = Fx, F = R B P (8) where P is positive symmetric matrix which is an unique solution of the Riccatti algebraic equation T 1 T PA A P PBR B P Q = (9) The closed-loop system is obtained by substituting (8) in (3), x = ( A BF )x (1) Note that, (6) can be viewed with the following relationship: T T T T ( A LC ) = A C L (11) Therefore, LQR can be applied from the duality relationship between the regulator and the observer by replacing (11) for (1). This replacement is done as follows: A T A, C T B, L T F (12) _ y As shown in (7), gain scheduling by tuneing the weighting matrices; greater Q means more control effort, and greater R results in conservative control input, in trade-off relationship. 3.3 Simulation results Parametric analysis is conducted for discussion of the turbine shaft parameter error. Generally, the parametric error from 1% to 2% between the actual value and the measurement value of turbine parameter. Moreover, the error of spring constant is larger than that of the inertia constant. Therefore, it is important to verify the observer s effectiveness under various errors. Fig.9 shows the actual six shaft torques obtained from simulation results. In addition, Fig.1 shows the estimated torque value for the following 4 cases, which uses the turbine parameter for the observer to comprise each error of measurement. The error combinations considered for inertia and spring constants respectively are (a): 5% and 1% (b): 1% and 15% (c): 1% and 3% (d): 3% and 4% The rotational speed is measured by the observation matrix C by which only the exciter side. On (a) and (b) of Fig.1, the torque estimation is excellently accomplished, the observer is applicapable for this kind of parametric error. However, the estimated amplitude especially after the disconnection of generator load mitigates in (c), this will result in poor performance for the application of the torque reduction control. Last figure (d) implies such large error gives wrong state estimation. Step1 Step2 Step3.7.6.5.4.3.2.1 -.1 -.2.1.2.3.4.5.6.7 Figure 9: Actual torsional torque.

.7.6.5.4 Estimated value (M5%, K1%).7.6.5.4 Estimated value (M1%, K15%).3.2.1.3.2.1 -.1 -.1 -.2.1.2.3.4.5.6.7.7 -.2.1.2.3.4.5.6.7 (a) inertia constant 5%, spring constant 1% (b) inertia constant 1%, spring constant 15% Estimated value (M2%, K3%).7 Estimated value (M3%, K4%).6.6.5.5.4.4.3.2.1.3.2.1 -.1 -.1 -.2.1.2.3.4.5.6.7 -.2.1.2.3.4.5.6.7 (c) inertia constant 1%, spring constant 3% (d) inertia constant 3%, spring constant 4% Figure 1: Estimated shaft torque. Fig.11 shows relation between the estimated shaft torque and parametric error. Figure (a) shows estimated torque when the error of spring constant is fix by 2% and the error of inertia constant is changed varied from 1% to 5% by 1%. Figure (a) shows estimated torque when the error of spring constant is fixed by 2% and the error of inertia constant is changed varied from 1% to 5% by 1%. Also figure (b) shows the torque when the error of inertia constant is fixed and that of spring constants is varied. The error of spring constant affects the estimated shaft torque more than that of inertia constant..7.6.5.4 M 1%, K 2% M 2%, K 2% M 3%, K 2% M 4%, K 2% M 5%, K 2%.6.5.4 M 2%, K 1% M 2%, K 2% M 2%, K 3% M 2%, K 4% M 2%, K 5%.3.2.1.3.2.1 -.1 -.1 -.2.1.2.3.4.5.6.7 (a) Effect of the error of inertia constant. Figure 11: Relation between estimation shaft torque and parametric error. -.2.1.2.3.4.5.6.7 (b) Effect of the error of spring constant.

4 OBSERVER BASED EXCITATION CONTROL In this chapter, the torque estimated by the observer is fed back into the excitation controller in order to reduce to the shaft torque. Fig. 12 shows the excitation control system for shaft torque reduction, where T obs is the estimated torque from the observer. The inertia constant error of the turbine parameter in this simulation is 1%, and the spring constant is 2%. Gain K determined by try and error. Generator's terminal Turbine speed 1. 1.5s Filter observer Tobs Reference K 1.2s 3.3 Setting value of 1. 1. 1.326s 1. Field voltag Brushless Figure 12: Excitation control system for shaft torque reduction. Fig. 13 illustrates the shaft torque with the shaft torque reduction excitation control. The shaft torque immediately after a fault could not be reduced. However, after disconnecting of the fault point, the torque is reduced effectively. Especially, after the general load is removed, the convergence time is greatly improved. The life cycle of the turbine shaft is degraded by accumulating fatigue. Therefore, it is effective to reduce the convergence time regarding main torque period..7.6.5.4.3.2.1 -.1 -.2.1.2.3.4.5.6.7.8.9 1 Figure 13: Shaft torque reduction with exciter control. 5 CONCLUSION This paper proposed an estimation method of the immeasurable shaft torque for exciter control using a torque observer. In the simulation results, it is clarified that the estimation accuracy is within about 9%. Also the estimation tolerance of the turbine parameter for observation is verified. Therefore, it is applicable for wide range operation. It is also shown that excitation control with the observer is effective for torque reduction. In future, this method will be applied for a thermal turbine generation system, but new counter measure may EF be required to eliminate errors caused by modal frequencies. Moreover, the most destructive torque at the fault cannot be prevented by the control method in this research. Therefore, this research will focus on the method of controlling the effective for the early torsional torque. REFERENCES [1] IEEE Torsional Issues Working Group, Fourth Supplement to a Bibliography for the Study of Subsynchronous Resonance Between Rotating Machines and Power Systems, IEEE Transactions on Power Systems, Vol.12, No.3, August 1997. [2] D. N. Walker, C. E. J. Bowler, and R. L. Jackson, Results of Subsynchronous Resonance at Mohave, IEEE Transactions on Power Apparatus and Systems, vol. PAS-94, No.5, September / October 1975. [3] D. N. Walker, S. L. Adams, and R. L. Jackson, Torsional Vibration and Fatigue of Turbine-Generator Shafts, IEEE Transactions on Power Apparatus and Systems, vol. PAS-1, No.11, November 1981. [4] J. S. Joyce, T. Kulig, and D. Lambrecht, Torsional Fatigue of Turbine-Generator Shafts Caused by Differential Electrical System Faults and Switching Operations, IEEE Transactions of Power Apparatus, Vol.PAS-97, September/October 1978. [5] Y.N.Yu, Electric Power System Dynamics, Academic Press Inc., London, 1983. [6] A. B. Abdennour, R. M. Hamouda, and A. A. Al-Ohaly, Countermeasures for Self-Excited Torsional Oscillations Using Reduced Order Robust Control Approach, IEEE Transactions on Power Systems, Vol.15., No.2, May 2. [7] C.E.J.Bowler, P.G.Brown, and D.N.Walker Evaluation of The Effect of Power Circuit Breaker Reclosing Practices on Turbine-Generator Shafts, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-99, No.5 Sept/Oct 198 [8] P.Kundur, Power System Stability and Control, EPRI Press, 1993. [9] P.M.Anderson, and A.A.Fouad, Power System Control and Stability, IEEE PRESS, 1993 [1] A.Abolins, D.Lambrecht, J.S.Joyce, and L.T.Rosenberg, Effect of Clearing Short Circuits and Automatic Reclosing on Torsional Stress and Life Expenditure of Turbine-Generator Shafts, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-95, No.1 January/February 1976 [11] T.Komukai, and M.Udo, Analysis of Transient Torques in Synchronous Machine at Multi-Phase Reclosing IEEE Transactions on Power Apparatus and Systems, Vol. PAS-92, No.1, pp.365-373 1973 [12] M. D. Brown, and C. Grande-Moran, Torsional System Parameter Identification of Turbine-Generator Sets, IEEE Transactions on Energy Conversion, Vol.12, No.4, December 2. [13] J. V. Milanovi, The Influence of Shaft Spring Constant Uncertainty on Torsional Modes of Turbogenerator, IEEE Transactions on Energy Conversion, Vol.13, No.2, June 1998.