Oscillation Damping With Optimal Pole-Shift Approach ... - IEEE Xplore
IEEE SYSTEMS JOURNAL, VOL. 3, NO. 3, SEPTEMBER 2009
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Oscillation Damping With Optimal Pole-Shift Approach in Application to a Hydro Plant Connected as SMIB System Nand Kishor, Member, IEEE
Abstract—This paper presents an application of control theory to damp out load angle and speed oscillations through the excitation and governor subsystems in a hydro power plant connected as single machine infinite bus (SMIB) system. The control technique used is based on optimal pole shift (PS) theory, unlike linear quadratic regulation (LQR) method, which requires solving an algebraic Riccati equation. The adopted approach offers satisfactory damping on speed and load angle oscillations. The proposed control scheme is demonstrated on light, normal and heavy load operating conditions. It is observed that with the proposed technique, the control design is robust over a wide range of operating conditions. Also in this paper, a comprehensive assessment of the effects of optimal control approach when applied with state-estimation techniques; linear Kalman filter (KF) and first-order divided difference filter (FDDF) has been addressed.
Constants of the linearized model of synchronous machine. Active and reactive power output of synchronous machine. Stabilizing transformer voltage.
,
Index Terms—Damping, estimation, optimal, pole-shift.
LIST OF SYMBOLS:
,
Available head at turbine inlet. Volumetric turbine flow.
,
, , ,
, ,
Water time constant of penstock , varies with load. Load disturbance. Developed turbine power. Gate opening. Blade position. Nonlinear function refereed to developed turbine power, flow. Damping torque.
,
, , , , ,
Mechanical starting time constant . Internal transient voltage in the -axis. -axis transient open circuit time constant. Terminal voltage. Transmission line impedance. Voltage regulator gain. Voltage regulator time constant . Stabilizing transformer gain, time constant.
Manuscript received October 08, 2008; revised February 04, 2009. First published July 21, 2009; current version published September 16, 2009. The author is with the Department of Electrical Engineering, Motilal Nehru National Institute of Technology, Allahabad-211004, India (e-mail: [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/JSYST.2009.2022577
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Field voltage. Reference voltage. Regulator voltage. Open-loop system state-space matrix. Control state-space matrix. Closed-loop system state-space matrix due to PS. Closed-loop system state-space matrix due to LQR. System control vector due to PS, LQR. System state vector. Estimation state vector due to KF, FDDF. Feedback gain matrix due to PS, LQR. Quadratic performance index due to PS, LQR. Number of dominant eigenvalues. Governor, Exciter output signal. Wicket gate, runner blade servomotor time constant. State weighting matrices corresponding to PS and LQR . Control weighting matrices, corresponding to PS and LQR . Positive real constant (scalar) to form image property. Open-loop eigenvalue, with real part , . imaginary part Closed-loop eigenvalue, with real part , . imaginary part Base angular speed (377.16 rad/s). Rotor angular speed. Load angle in radian (rad). I. INTRODUCTION
OWER plant oscillations occur due to the lack of damping torque at the generator rotor. The rotor oscillation causes the oscillation of power system variables (bus voltage, bus fre-
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Fig. 1. Hydro power plant connected as SMIB system.
quency, transmission line active, and reactive power, etc.). They are usually in the range between 0.1 to 2 Hz. This is a very important issue in power system operation and control for supplying sufficient and reliable electric power with good quality. In the classical design of controllers for the hydro turbinegenerator unit, the speed control and the excitation control were considered as two separate entities, which are independent to each other [1]. The reason is due to the fact that the operation of the speed loop is slower than the excitation loop. In recent years, digital adaptive control techniques have offered an enhanced performance in identification and control. The mutual coordination between the governor and the exciter control loop helps in obtaining better damping of transients and wider stability margins. Application of linear optimal regulation concept allows us to design a controller, which produces the best possible control system for a given set of performance objectives. The design of a control system is based on minimizing a quadratic performance index. The use of these approaches is discussed for power system applications and advantages over classical methods are well mentioned [2]–[5]. Herron et al. [2] formulated an observer-based controller using all states as input for hydro turbine control. The LQR/LQG [6] and LQG/LTR [7] approach for a state-space model is considered to control turbine speed. There has been some literatures [8]–[13] reported in which optimal PS approach has been applied in power system applications (power system stabilizer) and in the design of adaptive governor for synchronous operation of a hydroelectric unit [14]. A generalized multivariable PS adaptive control algorithm is presented in [8]. The technique provides on-line self-searching pole shifting factor to determine the excitation control limits over a wide operating range. An adaptive self-optimizing pole shift technique in the design of power system stabilizer is discussed in [9]. The control algorithm presented is based on combined essences of minimum variance and pole assignment. Furthermore, Chen et al. [10] have described an adaptive power system stabilizer. A recursive least-square method with a varying forgetting factor is used and amount of shift is determined by a PS factor. A state-feedback law based on PS approach is presented in [11]. The feedback gains and system parameters can be determined easily, as gains are linear functions of the PS factor. Radial basis function identifier with PS controller is applied for the design of power system
Fig. 2. Structural block diagram of hydro plant with controllers connected as SMIB system.
stabilizer in [12]. Abdelazim and Malik [13] have presented the Takagi-Sugeno (TS) fuzzy system to identify a synchronous machine model and the PS control is applied to calculate the control signal. A robust adaptive controller design using pole-shifting and parameter space method for governing of hydro turbine is described in [14]. Recently the pole shift control scheme based on mirror image property has been demonstrated for the power system stabilizer [15] and hydro power plant control [16]. This technique does not need the solution of nonlinear algebraic Riccati equation and easy to solve. In application to hydro plant few publications exist in which a part of the said approach has been adopted. However, the control design based on LQR requires measurement of all state variables, which is neither practical nor economical for most cases. Some state variable measurements in the stochastic plant model can be so noisy that a control system based on such measurements would be unreliable. Therefore, an estimator should be added to this design to estimate the immeasurable/noisy states. Such observerwouldnot takeinto account the power spectra of theprocess and measurement noise. These uncertainties can be accommodated to a large extent by LQ (linear quadratic) compensator. Estimator based on LQ approach for turbine control offers an alternative approach. Conventionally,a basic linear Kalman filter(KF) is seldomly used for system state estimation. A nonlinear extension of the same filter is extended Kalman filter (EKF). Until now, the EKF has been unquestionably considered as the dominating state estimation technique. A number of literatures are available in which its applications are mentioned. A derivative-free form
KISHOR: OSCILLATION DAMPING WITH OPTIMAL POLE-SHIFT APPROACH
of EKF is the first-order divided difference filter which uses Sterling’s interpolation formula to approximate a nonlinear function. An application of this approach to hydro power plant control design has been recently investigated in [17] wherein states were estimated for the compensation. The encouragement to work presented in this paper is obtained from the investigations carried out as discussed in [16], [17]. In the present paper, an enhancement in damping of oscillations in SMIB system through exciter and governor subsystems is presented. The control scheme based on optimal pole shift is employed [18]. Its solution is optimal with respect to a quadratic performance index, which can be easily found without solving any non linear algebraic Riccati equation. The time-domain, frequency-domain response and closed-loop eigenvalues are compared with those determined by LQR technique. The analysis is carried out to study the effectiveness of this technique under various load disturbances and plant operating conditions. The application of state feedback approach (LQR and PS) is also illustrated using the states load angle and angular speed estimated by linear KF and FDDF techniques as inputs.
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(4) The above partial derivatives vary with the operating point of the turbine, which in turn depend upon the load on the turbine/ remain constant for generator. The partial derivatives i.e., variation near the operating point. Their value depends on the initial steady-state point of the hydro turbine and they can be determined experimentally [21]. The hydraulic flow in the penstock is modeled with an assumption of in-elastic water column effect. The stiff water hammer equation from can be expressed as [22] (5) Simplifying (3) and (5) (6)
II. HYDRO PLANT CONNECTED AS SMIB SYSTEM A single Kaplan turbine-generator in a hydro power plant connected to local load and an infinite bus as shown in Fig. 1, is considered for the study. The linear model of SMIB system characterizes system oscillations accurately. This is due to the fact that system oscillations depend on the operating point of the power plant rather than the location and magnitude of the applied disturbance. The size and complexity of linear power plant model requires the use of state-space models. The usual approach to analyze these systems consists of computing the eigenvalues and eigenvectors of the state matrix. In the study, a state-space model with two-input and twooutput variables is considered. The control action is performed through the excitation and the governor subsystems. The dual regulation of hydro turbine is incorporated through the operation of both wicket gates and runner blades. The developed turand flow is a function of the utilized bine power , gate opening , speed , and blade position head . The relationship between these parameters is a nonlinear function written as [19], [20]
(1) (2) For small variations around an operating point the turbine parameters can be represented by the approximations through the Taylor series. In quite many literatures, state space realizations have been suggested to model hydro plant dynamics [19], [20]. For a given reference operating point, (1)–(2) may be modified and given in the partial derivative relationship between the variables as [19], [20]
The operation of the Kaplan turbine involves control of the wicket gates and the runner blades position in order to regulate the water flow to the hydro turbine. The corresponding servomotor equations are written as [19], [20] (7) (8) A third-order, model of synchronous generator is described by the following set of differential and algebraic equations [23]: (9) (10) (11) (12) The excitation system including its derivative type feedback compensation stabilization along with the automatic voltage regulator is modeled as [23] (13)
(14) (15) (16)
(3)
The dynamic characteristics of the system are expressed in terms of linearized constants defined in [23]:
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TABLE I QUANTITATIVE CHARACTERISTIC OF OPEN-LOOP EIGENVALUES AT VARIOUS OPERATING CONDITIONS
Fig. 4. Closed-loop Bode plot of output variables without load disturbance.
where the system state vector are defined as follows:
and the system control vector
(18) (19) with individual state variables, as indicated in the block-diagram of the complete SMIB system in Fig. 2. Here, is excitation is governor’s servomotor position. voltage signal and III. PROBLEM STATEMENT
Fig. 3. Dynamic response of output variables; tested with exponential signal for different load disturbances without controller at heavy load operating point.
The SMIB system represented in state-space form is given by the equation (17)
The enhancement in damping for the output variables can be defined as at a given operating condition, controller design stabilizes the system following a perturbation/disturbance. For the controller design to be considered “robust” enough, it must perform well for the other operating condition too, thus satisfying the design objectives. The application of optimal pole-shift in the hydro plant connected as SMIB system has been successfully demonstrated in [16]. In this paper, objective of the work is categorized into manifold. The emphasis of present study is to compare the wellknown linear quadratic regulation (LQR) approach with the op-
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Fig. 6. Pole-zero mapping for open loop and closed loop with PS control approach.
measurements. A number of researchers have demonstrated the use of linear KF, EKF, and more recently the derivative-free estimators like unscented Kalman filter, the central difference filter, and divided-difference filter (first-order and second-order approximations) [24]. Sections IV-A and B introduce on conventional linear KF and recent first-order divided-difference filter. The latter approach among the different filter variants has improved performance, compared to the extended Kalman filter, with the additional benefit in ease of implementation. The basic framework involves state estimation of the nonlinear dynamic system (20) (21)
Fig. 5. Bode plot of output variables; with optimal PS and LQR control approach at load disturbances.
timal pole-shift in damping out the oscillations at various op, erating conditions referred to as i) Heavy load: ; ii) Normal load: , ; and iii) Light load: , However, the control design based on LQR requires measurement of all state variables, which is neither practical nor viable on economical point of view. If the full-state of the system is not available, an estimator computes an estimate of the entire state vector when provided with the plant’s output. Thus, the next problem framed in study is the state vector estimation by linear Kalman filter and first-order divided difference filter approach and finally to combine the control law and estimator for plant compensation, wherein the control law calculations are based on the estimated states rather than the actual states of the SMIB system in order to dampen the state oscillations at various operating conditions. IV. STATE ESTIMATION TECHNIQUES This section addresses the problem of how to estimate the states of a SMIB system given the model structure and noisy
A. Linear Kalman Filter The basic conventional approach for state estimation is the linear Kalman filter (estimator). The solution of this estimator is a dual of LQR problem. The KF is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the plant states, in a way that minimizes the mean of the squared error. The filter is derived by computing an estimator for a linear state-space model of the SMIB system formulated in the form (22) (23) subject to additive white Gaussian noise that represents all these external error sources as included in (22)–(23). The KF addresses the general problem of trying to estimate the state of a controlled plant that is governed by . The linear stochastic (22)–(23) with a measurement KF assumes a plant that is subject to various disturbances and modeling errors. The internal states are determined in presence
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Fig. 7. Output variables—tested with exponential signal for different load disturbances with optimal PS and LQR control approach at heavy load operating point.
of plant noise and measurement noise. Define the plant noise covariance and measurement noise covariance as (24) In practice, the system noise covariance and measurement noise covariance matrices might change with time or measurement. It is often satisfactory to use a simplified time-invariant filter having a constant estimator gain
the states. The solution as discussed here for state estimation is a stochastic estimation algorithm. The FDDF is an improvement over the EKF that uses Sterling’s interpolation rather than the Taylor series derivative based approximation to linearize the system and observation processes. This estimation approach gives additional insight to performance compared to the others, with the added benefit of ease in implementation. The derivation of filter begins by replacing the linear system (22)–(23) with a general nonlinear model as
(25) with a priori estimate error covariance given as
(26) The state estimation problem can be easily solved using MATLAB toolbox lqe, which uses linear Kalman filter algorithm [25]. B. First-Order Divided Difference Filter (FDDF) In the state estimation of a nonlinear system especially as in a hydro plant, the basic linear Kalman filter overestimates
(27) (28) The FDDF uses Stirling’s interpolation formula to approximate a nonlinear function. Consider the operators and perform the following operations ( is defined interval length):
(29) (30)
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Fig. 8. Output variables—tested with step signal for different load disturbances with optimal PS and LQR control approach at light load operating point for gain matrix determined at heavy load operating point.
Sterling’s interpolation formula around the point terms of the above difference operators is [24]
in
where
The scalar form of the above interpolation formula is extended to the vector by introducing first-order divided difference oper, for a vector function ator (33) (31)
Then the first-order Taylor series expansion of about is given by [24]
First-order approximation of above polynomial takes the form (34) (32)
(38) (39) (40)
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TABLE II QUANTITATIVE ANALYSIS OF CLOSED-LOOP EIGENVALUES FOR CONTROLLER GAIN COMPUTED AT HEAVY LOAD OPERATING CONDITION
Then the first-order divided difference approximation of the function is given by
V. OPTIMAL REGULATION A. Linear Quadratic Regulation
(35) Next, the FDDF is applied to the expectation and covariance of vector of random variables, of probability distribution as [24] (36) (37) It is desired to estimate the following statistical measures for —see (38)–(40), as shown at the a nonlinear function of bottom of the previous page. The transformation matrix is selected as a square Cholesky factor of the covariance matrix
Linear optimal control and modern control theories introduced to improve the dynamic performance of systems depend on the accuracy of the model. Its reliability gets reduced as the system becomes larger and complex. In system modeling, the key problem is to find an appropriate model structure of a system with unknown parameters, given some prior knowledge about the system. The linear quadratic regulation measured by a quadratic performance criterion of form [17]
(44)
(43)
is minimized so as to transfer the system from its initial state to the final state. and are the state and control weighting matrices, respectively, determined by the designer to describe the relative importance to plant states and the cost of weight inputs more than resulting control. The value of the states while the value of weight the state more than the inputs. for linear quadratic design is given The feedback gain by
The details on the algorithm for state estimation by FDDF can be referred from [24].
(45)
(41) This leads to stochastic decoupling of the variables in
as (42)
The Estimator gain can be obtained as
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Fig. 9. Output variables—tested with step signal for different load disturbances with optimal PS and LQR control approach at light load operating point for gain matrix determined at normal load operating point.
TABLE III QUANTITATIVE ANALYSIS OF CLOSED-LOOP EIGENVALUES FOR CONTROL GAIN MATRIX COMPUTED AT NORMAL LOAD OPERATING CONDITION
where is the maximal solution of the continuous algebraic Riccati equation (CARE)
(46)
The closed-loop system matrix becomes (47) The regulation performance measured by a quadratic performance index (44) is minimized, i.e., the purpose of LQR is to determine the states and control signal so that (44) is minimum.
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TABLE IV QUANTITATIVE ANALYSIS OF CLOSED-LOOP EIGENVALUES FOR CONTROL GAIN MATRIX COMPUTED AT LIGHT LOAD OPERATING CONDITION
Minimization of results to bring the states minimum control energy and state deviations.
to zero with
B. Optimal-Pole Shift In the present work, optimal PS approach as suggested in [18] is utilized. The method does not require solving nonlinear algebraic Riccati equation. Just a first-order or a second-order linear Lyapunov equation is to be solved for shifting one real pole or two complex conjugate poles, respectively. In shift of complex conjugate poles to a suitable location for a desired damping coefficient, only real part of the open-loop complex conjugate poles are shifted keeping the magnitude of imaginary parts preserved. The shift of real/complex conjugate poles leads to an optimal feedback control law with respect to a quadratic performance index [18]. For a controllable system given as (48) it is required to design a feedback matrix which shifts the to the desired locations real parts of the open-loop poles ; while the imaginary parts are kept preserved. At the same time should minimize a quadratic performance index given by (49) Here the closed-loop system matrix is defined as (50) The application of the above approach has been successfully demonstrated in [16] but the comparison with well known LQR technique is not presented.
VI. RESULTS AND DISCUSSION The hydro plant connected as SMIB with local load at generator terminals is represented as two-input and two-output system. The two inputs are exciter voltage and gate-blade signal and the two outputs are angular speed and load angle. The parameters considered in the study for the power plant is given , [16]. At operating condition: (heavy load), , (normal load) and , (light load), open-loop are determined from the (17). Table I depicts eigenvalues the open-loop eigenvalues of the system under study for various operating conditions. The values of damping coefficient and undamped natural frequency corresponding to each open-loop eigenvalues is represented in square bracket. The output time response of the plant without controller with exponential change (load disturbance): 0.02 and 0.05 p.u. increment and 0.1 p.u. reduction from initial operating point is illustrated in Fig. 3. From these figures, the output response without controller is observed to be highly oscillatory, over considerable time duration. A. State Feedback Controller Performance In the study, optimal PS approach is first applied at heavy load operating condition. The dominant eigenvalues (i.e., located nearer to imaginary axis) are shifted to new position on the left-hand side in s-plane (preserving the imaginary component) so as to damp out with desired damping coefficient the oscillation. And depending upon the real or complex poles is computed assuming to be shifted, the controller gain . , the Riccati equation And to compute gain matrix is solved using MATLAB® control toolbox [25]. The state is assumed as weighting matrix
KISHOR: OSCILLATION DAMPING WITH OPTIMAL POLE-SHIFT APPROACH
Bode plot study without load disturbance and with load disturbance at heavy load operating condition is shown in Figs. 4 and 5, respectively. It is indicated that, the Bode plot due to both PS and LQR techniques overlap each other and reflect a similar behavior. The representation of poles in open loop and closed loop with PS control approach is illustrated in Fig. 6. Further studies to examine the effectiveness of the control approach, the time domain simulations are performed on the SMIB system. The closed-loop time response simulated with exponential change in load disturbances as stated above are shown in Fig. 7. The simulated response in Fig. 7(a) illustrate that the optimal PS control method bring about considerable damping effect on the load angle oscillation with respect to Fig. 3(a). It is indicated that swing of the output variables has, being damped, reduced overshoot and settling time. Although both the techniques provide quite satisfactory performance, however, PS yields a faster settling time of the dynamics, along with smaller peak deviations. However, speed response due to PS and LQR techniques are of similar characteristic as shown in Fig. 7(b). To achieve clear illustration, the zoomed response for 0.05 p.u. increment in load (disturbance) with both LQR and PS control approach are also presented. The changes in operating conditions of the nonlinear system can be considered as perturbations in the coefficients of matrices for linearized system. The simultaneous stabilization of the output variables is demonstrated at other loading conditions via a single setting of controller gain matrix in the following paragraphs. at operHaving determined the controller gain matrix , , closed-loop ating condition, eigenvalues are determined at initial operating points: , (normal load) and , (light load) with substitution of controller gain matrix in respective closed-loop system matrix (50). Similarly, determined at heavy load operthe controller gain matrix ating condition is substituted in matrix (47) of normal and light load operating points to compute closed-loop eigenvalues. These eigenvalues are given in Table II. It is found that the optimal PS leads to a shift of dominant open-loop eigenvalues at other operating points too. The closed-loop time response simulated with step signal at above stated conditions are shown in Fig. 8. A similar conclusion can be drawn about these as discussed for Fig. 7. The performance of PS is equally good and, in some cases, better when compared to a LQR approach. To further ascertain the feasibility of the optimal PS techis determined at normal load operating nique, similarly condition after shift of dominant open-loop eigenvalues to new location. Then closed-loop eigenvalues are evaluated at heavy and light load operating condition to verify the shift of dominant eigenvalues. The closed-loop time response simulated with step signal at light load operating point for gain matrix determined at normal load operating point is shown in Fig. 9. It is noted that the output response reaches towards the zero value with the short time-duration scale. The overshoot/undershot amplitude variation of output variables with PS approach is comparatively lower as indicated in Fig. 9(a). The speed deviation response so obtained is exactly similar as shown in Fig. 9(b). The action of controllers provides satisfactory response for speed de-
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viation overshoot and transient oscillation with zero steady state error. Table III presents the closed-loop eigenvalues at normal load operating point and the same evaluated at heavy and light load operating point to verify the shift of dominant ones. Similar study is performed with gain matrix determined at light load operating point. Table IV presents the closed-loop eigenvalues determined at light load operating point. The observation suggests that location of closed-loop eigenvalues obtained from LQR method does not differ much with changes in operating points. It is also observed that with optimal PS approach; only dominant eigenvalues get shifted to new optimal location irrespective of any subsequent change in operating point. Thus, the control gain matrix determined using optimal PS approach is independent of system operating conditions, i.e., its variation do not adversely affect the controller performance. It is possible to select a single set controller gain matrix to ensure the stabilization of the system over a wide an operating range. Thus from the above discussion it is understood, optimal PS performance is as good as widely used LQR, with an advantage to ease in computation of control gain matrix. B. Estimator Based Controller Performance Linear optimal control and modern control theories introduced to improve the dynamic performance of systems however depend on the accuracy of the model. Its reliability gets reduced as the system becomes larger and complex. In use of state-feedback theories, the states are assumed to be measurable or are estimated. The following discuss the estimation of the states of SMIB system with noisy measurements. The process noise and measurement noise are considered to be uncorrelated and unbiased. The state (12)–(19) are solved using a fourth-order Runge–Kutta method with 64 steps taken between each observation. The closed-loop initial response of the compensated SMIB system with LQR and PS approach is shown in Fig. 10. The estimated states and state-error are shown in these figures. The observation is Fig. 10(a) suggests that the initial response of estimated states using LQR and PS control approach yield similar variation. Fig. 10(b) also illustrates the same finding. Further the comparison between the estimation techniques as shown in Fig. 10(a)–(b), it is understood that the state-error (error LQR and error PS) due to linear KF results in comparatively higher overshoot/undershoot, but with faster settling time. The KF is unable to completely estimate the state due to time delay problems and unmodeled dynamic problems and thus tends to over compensate. The comparison of control effort required in each estimation techniques is presented in Fig. 11. The control signals in case of states estimated by linear KF vary over a comparatively large magnitude but settles down quickly. This is a vivid example of the classical approach trade-off between response time and overshoot. In practice, a minimum control effort is desired to achieve sufficient damping of controlled signals. Furthermore, for quantitative analysis, the measure of state estimation techniques and optimal controller has been considered by defining the performance index as shown in the equation at the bottom of the next page, where defines the steadystate values following the initial condition test. The minimization of these values would provide improved dynamic perfor-
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Fig. 10. Estimated states and state-error variation with optimal control approach at heavy load operating condition.
Fig. 11. Control signal variation with optimal control approach at heavy load operating condition.
mance system and the computed values are shown in Fig. 12. The compensation by both PS and LQR using FDDF technique (i.e., PS-FDDF, LQR-FDDF) presents superior states estimation. This is concluded from the bar-chart representation of performance index as shown in Fig. 12(a). The quantitative analysis for required control effort to support the FDDF estimation technique is illustrated in Fig. 12(b). At normal/light load conditions, the computed control signal (PS-FDDF & LQR-FDDF))
becomes comparatively too small to be represented as a barchart. VII. CONCLUSION The hydro power plant control as SMIB system was studied in the paper. An approach towards achieving an optimal control using pole shift satisfying a quadratic performance index was
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Fig. 12. Bar-chart representation of performance index for state estimation techniques and optimal controller.
discussed. In this study, the damping of load angle and speed deviation via LQR and PS, when applied independently and also through state estimator was investigated and discussed for different operating conditions. The time response and frequency response results illustrated that the said method was effective to damp out the load angle and speed oscillations. It suggested a response as equivalent to linear quadratic regulator and in some cases better with an advantage to ease in computation of controller gain matrix. The simultaneous stabilization of the SMIB system was demonstrated by considering three different operating conditions as heavy load, normal load, and light load. The controller gain matrix determined at any operating point ensured shift of dominant open-loop eigenvalues at other operating points too. Thus, a control scheme provided great damping characteristics and enhanced significantly the system stability with even changes in plant operating conditions. Briefly, the system remained robust over a wide range of change in operating conditions. The state-feedback control approach is further illustrated with estimation of states using linear Kalman filter and first-order divided difference filter. The state-error based on linear Kalman filter estimation results in comparatively higher overshoot/undershoot, but with faster settling time. The comparison between the estimation techniques suggested that the FDDF provides a comparatively lower control effort but at expense of higher oscillation before settling down.
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[11] M. L. Kothari, A. K. Bhattacharya, and J. Nanda, “Adaptive power system stabilizer based on pole-shifting technique,” IEE Proc. Gen., Trans., and Distrib., vol. 143, pp. 96–98, 1996. [12] G. Ramakrishna and O. P. Malik, “RBF identifier and pole shifting controller for PSS application,” in Proc. Int. Conf. Electric Machines and Drives (IEMD’ 99), 1999, pp. 589–591. [13] T. Abdelazim and O. P. Malik, “Fuzzy logic based identifier and poleshifting controller for PSS application,” in Proc. IEEE Power Eng. Soc. General Meeting, 2003, pp. 1680–1685. [14] Y. Zeng and O. P. Malik, “Robust adaptive controller design based on pole-shifting technique,” in Proc. 32nd Conf. Decision and Control, 1993, pp. 2353–2357. [15] M. K. El-Sherbiny, M. M. Hasan, G. El-Saady, and A. M. Yousef, “Optimal pole shifting for power system stabilization,” Elect. Power Syst. Res., vol. 66, pp. 253–258, 2003. [16] N. Kishor, R. P. Saini, and S. P. Singh, “Optimal pole shift control in application to a hydro power plant,” J. Elect. Eng., vol. 56, pp. 290–297, 2005. [17] N. Kishor, S. P. Singh, and A. S. Raghuvanshi, “Dynamic simulation of hydro turbine and its state estimation based LQ control,” Energy Convers. Manag., vol. 47, pp. 3119–3137, 2006. [18] M. H. Amin, “Optimal pole shifting for continuous multivariable linear systems,” Int. J. Control, vol. 41, pp. 701–707, 1995. [19] M. B. Djukanovic, M. S. Calovic, B. Vesovic, and D. J. Sobajic, “Neuro-fuzzy controller of low head power plants using adaptive-network based fuzzy inference system,” IEEE Trans. Energy Convers., vol. 12, no. 4, pp. 375–381, Dec. 1997. [20] M. B. Djukanovic, D. J. Dobrijevic, M. S. Calovic, M. Novicevic, and D. J. Sobajic, “Coordinated stabilizing control for the exciter and governor loops using fuzzy set theory and neural nets,” Elect. Power Energy Syst., vol. 19, pp. 489–499, 1997.
[21] S. Mansoor, “Behaviour and Operation of Pumped Storage of Hydro Plants,” Ph.D. dissertation, University of Wales, Bangor, U.K., 2000. [22] P. Kundur, Power System Stability and Control. New York: EPRI/ McGraw-Hill, 1994. [23] P. M. Anderson and A. A. Fouad, Power System Control and Stability. New York: IEEE Press/Wiley, 2003. [24] M. Nørgaard, N. K. Poulsen, and O. Ravn, “New developments in state estimation for nonlinear systems,” Automatica, vol. 36, pp. 1627–1638, 2000. [25] Control System Toolbox, MATLAB® The Math Works, Inc., Natick, MA, 2000.
Nand Kishor (SM’04–AM’06–M’07) received the Ph.D. degree from the Indian Institute of Technology, Roorkee, in 2006. He is an Assistant Professor in the Department of Electrical Engineering, Motilal Nehru National Institute of Technology, Allahabad, India. His main research includes power plant modeling and control, application of artificial intelligence in system modeling, and renewable energy systems. He has published his research in reputed international referred journals and proceedings and is a regular reviewer of manuscripts from referred journals of well-known publishers.
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Oscillation Damping With Optimal Pole-Shift Approach in Application to a Hydro Plant Connected as SMIB System Nand Kishor, Member, IEEE
Abstract—This paper presents an application of control theory to damp out load angle and speed oscillations through the excitation and governor subsystems in a hydro power plant connected as single machine infinite bus (SMIB) system. The control technique used is based on optimal pole shift (PS) theory, unlike linear quadratic regulation (LQR) method, which requires solving an algebraic Riccati equation. The adopted approach offers satisfactory damping on speed and load angle oscillations. The proposed control scheme is demonstrated on light, normal and heavy load operating conditions. It is observed that with the proposed technique, the control design is robust over a wide range of operating conditions. Also in this paper, a comprehensive assessment of the effects of optimal control approach when applied with state-estimation techniques; linear Kalman filter (KF) and first-order divided difference filter (FDDF) has been addressed.
Constants of the linearized model of synchronous machine. Active and reactive power output of synchronous machine. Stabilizing transformer voltage.
,
Index Terms—Damping, estimation, optimal, pole-shift.
LIST OF SYMBOLS:
,
Available head at turbine inlet. Volumetric turbine flow.
,
, , ,
, ,
Water time constant of penstock , varies with load. Load disturbance. Developed turbine power. Gate opening. Blade position. Nonlinear function refereed to developed turbine power, flow. Damping torque.
,
, , , , ,
Mechanical starting time constant . Internal transient voltage in the -axis. -axis transient open circuit time constant. Terminal voltage. Transmission line impedance. Voltage regulator gain. Voltage regulator time constant . Stabilizing transformer gain, time constant.
Manuscript received October 08, 2008; revised February 04, 2009. First published July 21, 2009; current version published September 16, 2009. The author is with the Department of Electrical Engineering, Motilal Nehru National Institute of Technology, Allahabad-211004, India (e-mail: [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/JSYST.2009.2022577
,
Field voltage. Reference voltage. Regulator voltage. Open-loop system state-space matrix. Control state-space matrix. Closed-loop system state-space matrix due to PS. Closed-loop system state-space matrix due to LQR. System control vector due to PS, LQR. System state vector. Estimation state vector due to KF, FDDF. Feedback gain matrix due to PS, LQR. Quadratic performance index due to PS, LQR. Number of dominant eigenvalues. Governor, Exciter output signal. Wicket gate, runner blade servomotor time constant. State weighting matrices corresponding to PS and LQR . Control weighting matrices, corresponding to PS and LQR . Positive real constant (scalar) to form image property. Open-loop eigenvalue, with real part , . imaginary part Closed-loop eigenvalue, with real part , . imaginary part Base angular speed (377.16 rad/s). Rotor angular speed. Load angle in radian (rad). I. INTRODUCTION
OWER plant oscillations occur due to the lack of damping torque at the generator rotor. The rotor oscillation causes the oscillation of power system variables (bus voltage, bus fre-
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Fig. 1. Hydro power plant connected as SMIB system.
quency, transmission line active, and reactive power, etc.). They are usually in the range between 0.1 to 2 Hz. This is a very important issue in power system operation and control for supplying sufficient and reliable electric power with good quality. In the classical design of controllers for the hydro turbinegenerator unit, the speed control and the excitation control were considered as two separate entities, which are independent to each other [1]. The reason is due to the fact that the operation of the speed loop is slower than the excitation loop. In recent years, digital adaptive control techniques have offered an enhanced performance in identification and control. The mutual coordination between the governor and the exciter control loop helps in obtaining better damping of transients and wider stability margins. Application of linear optimal regulation concept allows us to design a controller, which produces the best possible control system for a given set of performance objectives. The design of a control system is based on minimizing a quadratic performance index. The use of these approaches is discussed for power system applications and advantages over classical methods are well mentioned [2]–[5]. Herron et al. [2] formulated an observer-based controller using all states as input for hydro turbine control. The LQR/LQG [6] and LQG/LTR [7] approach for a state-space model is considered to control turbine speed. There has been some literatures [8]–[13] reported in which optimal PS approach has been applied in power system applications (power system stabilizer) and in the design of adaptive governor for synchronous operation of a hydroelectric unit [14]. A generalized multivariable PS adaptive control algorithm is presented in [8]. The technique provides on-line self-searching pole shifting factor to determine the excitation control limits over a wide operating range. An adaptive self-optimizing pole shift technique in the design of power system stabilizer is discussed in [9]. The control algorithm presented is based on combined essences of minimum variance and pole assignment. Furthermore, Chen et al. [10] have described an adaptive power system stabilizer. A recursive least-square method with a varying forgetting factor is used and amount of shift is determined by a PS factor. A state-feedback law based on PS approach is presented in [11]. The feedback gains and system parameters can be determined easily, as gains are linear functions of the PS factor. Radial basis function identifier with PS controller is applied for the design of power system
Fig. 2. Structural block diagram of hydro plant with controllers connected as SMIB system.
stabilizer in [12]. Abdelazim and Malik [13] have presented the Takagi-Sugeno (TS) fuzzy system to identify a synchronous machine model and the PS control is applied to calculate the control signal. A robust adaptive controller design using pole-shifting and parameter space method for governing of hydro turbine is described in [14]. Recently the pole shift control scheme based on mirror image property has been demonstrated for the power system stabilizer [15] and hydro power plant control [16]. This technique does not need the solution of nonlinear algebraic Riccati equation and easy to solve. In application to hydro plant few publications exist in which a part of the said approach has been adopted. However, the control design based on LQR requires measurement of all state variables, which is neither practical nor economical for most cases. Some state variable measurements in the stochastic plant model can be so noisy that a control system based on such measurements would be unreliable. Therefore, an estimator should be added to this design to estimate the immeasurable/noisy states. Such observerwouldnot takeinto account the power spectra of theprocess and measurement noise. These uncertainties can be accommodated to a large extent by LQ (linear quadratic) compensator. Estimator based on LQ approach for turbine control offers an alternative approach. Conventionally,a basic linear Kalman filter(KF) is seldomly used for system state estimation. A nonlinear extension of the same filter is extended Kalman filter (EKF). Until now, the EKF has been unquestionably considered as the dominating state estimation technique. A number of literatures are available in which its applications are mentioned. A derivative-free form
KISHOR: OSCILLATION DAMPING WITH OPTIMAL POLE-SHIFT APPROACH
of EKF is the first-order divided difference filter which uses Sterling’s interpolation formula to approximate a nonlinear function. An application of this approach to hydro power plant control design has been recently investigated in [17] wherein states were estimated for the compensation. The encouragement to work presented in this paper is obtained from the investigations carried out as discussed in [16], [17]. In the present paper, an enhancement in damping of oscillations in SMIB system through exciter and governor subsystems is presented. The control scheme based on optimal pole shift is employed [18]. Its solution is optimal with respect to a quadratic performance index, which can be easily found without solving any non linear algebraic Riccati equation. The time-domain, frequency-domain response and closed-loop eigenvalues are compared with those determined by LQR technique. The analysis is carried out to study the effectiveness of this technique under various load disturbances and plant operating conditions. The application of state feedback approach (LQR and PS) is also illustrated using the states load angle and angular speed estimated by linear KF and FDDF techniques as inputs.
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(4) The above partial derivatives vary with the operating point of the turbine, which in turn depend upon the load on the turbine/ remain constant for generator. The partial derivatives i.e., variation near the operating point. Their value depends on the initial steady-state point of the hydro turbine and they can be determined experimentally [21]. The hydraulic flow in the penstock is modeled with an assumption of in-elastic water column effect. The stiff water hammer equation from can be expressed as [22] (5) Simplifying (3) and (5) (6)
II. HYDRO PLANT CONNECTED AS SMIB SYSTEM A single Kaplan turbine-generator in a hydro power plant connected to local load and an infinite bus as shown in Fig. 1, is considered for the study. The linear model of SMIB system characterizes system oscillations accurately. This is due to the fact that system oscillations depend on the operating point of the power plant rather than the location and magnitude of the applied disturbance. The size and complexity of linear power plant model requires the use of state-space models. The usual approach to analyze these systems consists of computing the eigenvalues and eigenvectors of the state matrix. In the study, a state-space model with two-input and twooutput variables is considered. The control action is performed through the excitation and the governor subsystems. The dual regulation of hydro turbine is incorporated through the operation of both wicket gates and runner blades. The developed turand flow is a function of the utilized bine power , gate opening , speed , and blade position head . The relationship between these parameters is a nonlinear function written as [19], [20]
(1) (2) For small variations around an operating point the turbine parameters can be represented by the approximations through the Taylor series. In quite many literatures, state space realizations have been suggested to model hydro plant dynamics [19], [20]. For a given reference operating point, (1)–(2) may be modified and given in the partial derivative relationship between the variables as [19], [20]
The operation of the Kaplan turbine involves control of the wicket gates and the runner blades position in order to regulate the water flow to the hydro turbine. The corresponding servomotor equations are written as [19], [20] (7) (8) A third-order, model of synchronous generator is described by the following set of differential and algebraic equations [23]: (9) (10) (11) (12) The excitation system including its derivative type feedback compensation stabilization along with the automatic voltage regulator is modeled as [23] (13)
(14) (15) (16)
(3)
The dynamic characteristics of the system are expressed in terms of linearized constants defined in [23]:
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TABLE I QUANTITATIVE CHARACTERISTIC OF OPEN-LOOP EIGENVALUES AT VARIOUS OPERATING CONDITIONS
Fig. 4. Closed-loop Bode plot of output variables without load disturbance.
where the system state vector are defined as follows:
and the system control vector
(18) (19) with individual state variables, as indicated in the block-diagram of the complete SMIB system in Fig. 2. Here, is excitation is governor’s servomotor position. voltage signal and III. PROBLEM STATEMENT
Fig. 3. Dynamic response of output variables; tested with exponential signal for different load disturbances without controller at heavy load operating point.
The SMIB system represented in state-space form is given by the equation (17)
The enhancement in damping for the output variables can be defined as at a given operating condition, controller design stabilizes the system following a perturbation/disturbance. For the controller design to be considered “robust” enough, it must perform well for the other operating condition too, thus satisfying the design objectives. The application of optimal pole-shift in the hydro plant connected as SMIB system has been successfully demonstrated in [16]. In this paper, objective of the work is categorized into manifold. The emphasis of present study is to compare the wellknown linear quadratic regulation (LQR) approach with the op-
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Fig. 6. Pole-zero mapping for open loop and closed loop with PS control approach.
measurements. A number of researchers have demonstrated the use of linear KF, EKF, and more recently the derivative-free estimators like unscented Kalman filter, the central difference filter, and divided-difference filter (first-order and second-order approximations) [24]. Sections IV-A and B introduce on conventional linear KF and recent first-order divided-difference filter. The latter approach among the different filter variants has improved performance, compared to the extended Kalman filter, with the additional benefit in ease of implementation. The basic framework involves state estimation of the nonlinear dynamic system (20) (21)
Fig. 5. Bode plot of output variables; with optimal PS and LQR control approach at load disturbances.
timal pole-shift in damping out the oscillations at various op, erating conditions referred to as i) Heavy load: ; ii) Normal load: , ; and iii) Light load: , However, the control design based on LQR requires measurement of all state variables, which is neither practical nor viable on economical point of view. If the full-state of the system is not available, an estimator computes an estimate of the entire state vector when provided with the plant’s output. Thus, the next problem framed in study is the state vector estimation by linear Kalman filter and first-order divided difference filter approach and finally to combine the control law and estimator for plant compensation, wherein the control law calculations are based on the estimated states rather than the actual states of the SMIB system in order to dampen the state oscillations at various operating conditions. IV. STATE ESTIMATION TECHNIQUES This section addresses the problem of how to estimate the states of a SMIB system given the model structure and noisy
A. Linear Kalman Filter The basic conventional approach for state estimation is the linear Kalman filter (estimator). The solution of this estimator is a dual of LQR problem. The KF is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the plant states, in a way that minimizes the mean of the squared error. The filter is derived by computing an estimator for a linear state-space model of the SMIB system formulated in the form (22) (23) subject to additive white Gaussian noise that represents all these external error sources as included in (22)–(23). The KF addresses the general problem of trying to estimate the state of a controlled plant that is governed by . The linear stochastic (22)–(23) with a measurement KF assumes a plant that is subject to various disturbances and modeling errors. The internal states are determined in presence
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Fig. 7. Output variables—tested with exponential signal for different load disturbances with optimal PS and LQR control approach at heavy load operating point.
of plant noise and measurement noise. Define the plant noise covariance and measurement noise covariance as (24) In practice, the system noise covariance and measurement noise covariance matrices might change with time or measurement. It is often satisfactory to use a simplified time-invariant filter having a constant estimator gain
the states. The solution as discussed here for state estimation is a stochastic estimation algorithm. The FDDF is an improvement over the EKF that uses Sterling’s interpolation rather than the Taylor series derivative based approximation to linearize the system and observation processes. This estimation approach gives additional insight to performance compared to the others, with the added benefit of ease in implementation. The derivation of filter begins by replacing the linear system (22)–(23) with a general nonlinear model as
(25) with a priori estimate error covariance given as
(26) The state estimation problem can be easily solved using MATLAB toolbox lqe, which uses linear Kalman filter algorithm [25]. B. First-Order Divided Difference Filter (FDDF) In the state estimation of a nonlinear system especially as in a hydro plant, the basic linear Kalman filter overestimates
(27) (28) The FDDF uses Stirling’s interpolation formula to approximate a nonlinear function. Consider the operators and perform the following operations ( is defined interval length):
(29) (30)
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Fig. 8. Output variables—tested with step signal for different load disturbances with optimal PS and LQR control approach at light load operating point for gain matrix determined at heavy load operating point.
Sterling’s interpolation formula around the point terms of the above difference operators is [24]
in
where
The scalar form of the above interpolation formula is extended to the vector by introducing first-order divided difference oper, for a vector function ator (33) (31)
Then the first-order Taylor series expansion of about is given by [24]
First-order approximation of above polynomial takes the form (34) (32)
(38) (39) (40)
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TABLE II QUANTITATIVE ANALYSIS OF CLOSED-LOOP EIGENVALUES FOR CONTROLLER GAIN COMPUTED AT HEAVY LOAD OPERATING CONDITION
Then the first-order divided difference approximation of the function is given by
V. OPTIMAL REGULATION A. Linear Quadratic Regulation
(35) Next, the FDDF is applied to the expectation and covariance of vector of random variables, of probability distribution as [24] (36) (37) It is desired to estimate the following statistical measures for —see (38)–(40), as shown at the a nonlinear function of bottom of the previous page. The transformation matrix is selected as a square Cholesky factor of the covariance matrix
Linear optimal control and modern control theories introduced to improve the dynamic performance of systems depend on the accuracy of the model. Its reliability gets reduced as the system becomes larger and complex. In system modeling, the key problem is to find an appropriate model structure of a system with unknown parameters, given some prior knowledge about the system. The linear quadratic regulation measured by a quadratic performance criterion of form [17]
(44)
(43)
is minimized so as to transfer the system from its initial state to the final state. and are the state and control weighting matrices, respectively, determined by the designer to describe the relative importance to plant states and the cost of weight inputs more than resulting control. The value of the states while the value of weight the state more than the inputs. for linear quadratic design is given The feedback gain by
The details on the algorithm for state estimation by FDDF can be referred from [24].
(45)
(41) This leads to stochastic decoupling of the variables in
as (42)
The Estimator gain can be obtained as
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Fig. 9. Output variables—tested with step signal for different load disturbances with optimal PS and LQR control approach at light load operating point for gain matrix determined at normal load operating point.
TABLE III QUANTITATIVE ANALYSIS OF CLOSED-LOOP EIGENVALUES FOR CONTROL GAIN MATRIX COMPUTED AT NORMAL LOAD OPERATING CONDITION
where is the maximal solution of the continuous algebraic Riccati equation (CARE)
(46)
The closed-loop system matrix becomes (47) The regulation performance measured by a quadratic performance index (44) is minimized, i.e., the purpose of LQR is to determine the states and control signal so that (44) is minimum.
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TABLE IV QUANTITATIVE ANALYSIS OF CLOSED-LOOP EIGENVALUES FOR CONTROL GAIN MATRIX COMPUTED AT LIGHT LOAD OPERATING CONDITION
Minimization of results to bring the states minimum control energy and state deviations.
to zero with
B. Optimal-Pole Shift In the present work, optimal PS approach as suggested in [18] is utilized. The method does not require solving nonlinear algebraic Riccati equation. Just a first-order or a second-order linear Lyapunov equation is to be solved for shifting one real pole or two complex conjugate poles, respectively. In shift of complex conjugate poles to a suitable location for a desired damping coefficient, only real part of the open-loop complex conjugate poles are shifted keeping the magnitude of imaginary parts preserved. The shift of real/complex conjugate poles leads to an optimal feedback control law with respect to a quadratic performance index [18]. For a controllable system given as (48) it is required to design a feedback matrix which shifts the to the desired locations real parts of the open-loop poles ; while the imaginary parts are kept preserved. At the same time should minimize a quadratic performance index given by (49) Here the closed-loop system matrix is defined as (50) The application of the above approach has been successfully demonstrated in [16] but the comparison with well known LQR technique is not presented.
VI. RESULTS AND DISCUSSION The hydro plant connected as SMIB with local load at generator terminals is represented as two-input and two-output system. The two inputs are exciter voltage and gate-blade signal and the two outputs are angular speed and load angle. The parameters considered in the study for the power plant is given , [16]. At operating condition: (heavy load), , (normal load) and , (light load), open-loop are determined from the (17). Table I depicts eigenvalues the open-loop eigenvalues of the system under study for various operating conditions. The values of damping coefficient and undamped natural frequency corresponding to each open-loop eigenvalues is represented in square bracket. The output time response of the plant without controller with exponential change (load disturbance): 0.02 and 0.05 p.u. increment and 0.1 p.u. reduction from initial operating point is illustrated in Fig. 3. From these figures, the output response without controller is observed to be highly oscillatory, over considerable time duration. A. State Feedback Controller Performance In the study, optimal PS approach is first applied at heavy load operating condition. The dominant eigenvalues (i.e., located nearer to imaginary axis) are shifted to new position on the left-hand side in s-plane (preserving the imaginary component) so as to damp out with desired damping coefficient the oscillation. And depending upon the real or complex poles is computed assuming to be shifted, the controller gain . , the Riccati equation And to compute gain matrix is solved using MATLAB® control toolbox [25]. The state is assumed as weighting matrix
KISHOR: OSCILLATION DAMPING WITH OPTIMAL POLE-SHIFT APPROACH
Bode plot study without load disturbance and with load disturbance at heavy load operating condition is shown in Figs. 4 and 5, respectively. It is indicated that, the Bode plot due to both PS and LQR techniques overlap each other and reflect a similar behavior. The representation of poles in open loop and closed loop with PS control approach is illustrated in Fig. 6. Further studies to examine the effectiveness of the control approach, the time domain simulations are performed on the SMIB system. The closed-loop time response simulated with exponential change in load disturbances as stated above are shown in Fig. 7. The simulated response in Fig. 7(a) illustrate that the optimal PS control method bring about considerable damping effect on the load angle oscillation with respect to Fig. 3(a). It is indicated that swing of the output variables has, being damped, reduced overshoot and settling time. Although both the techniques provide quite satisfactory performance, however, PS yields a faster settling time of the dynamics, along with smaller peak deviations. However, speed response due to PS and LQR techniques are of similar characteristic as shown in Fig. 7(b). To achieve clear illustration, the zoomed response for 0.05 p.u. increment in load (disturbance) with both LQR and PS control approach are also presented. The changes in operating conditions of the nonlinear system can be considered as perturbations in the coefficients of matrices for linearized system. The simultaneous stabilization of the output variables is demonstrated at other loading conditions via a single setting of controller gain matrix in the following paragraphs. at operHaving determined the controller gain matrix , , closed-loop ating condition, eigenvalues are determined at initial operating points: , (normal load) and , (light load) with substitution of controller gain matrix in respective closed-loop system matrix (50). Similarly, determined at heavy load operthe controller gain matrix ating condition is substituted in matrix (47) of normal and light load operating points to compute closed-loop eigenvalues. These eigenvalues are given in Table II. It is found that the optimal PS leads to a shift of dominant open-loop eigenvalues at other operating points too. The closed-loop time response simulated with step signal at above stated conditions are shown in Fig. 8. A similar conclusion can be drawn about these as discussed for Fig. 7. The performance of PS is equally good and, in some cases, better when compared to a LQR approach. To further ascertain the feasibility of the optimal PS techis determined at normal load operating nique, similarly condition after shift of dominant open-loop eigenvalues to new location. Then closed-loop eigenvalues are evaluated at heavy and light load operating condition to verify the shift of dominant eigenvalues. The closed-loop time response simulated with step signal at light load operating point for gain matrix determined at normal load operating point is shown in Fig. 9. It is noted that the output response reaches towards the zero value with the short time-duration scale. The overshoot/undershot amplitude variation of output variables with PS approach is comparatively lower as indicated in Fig. 9(a). The speed deviation response so obtained is exactly similar as shown in Fig. 9(b). The action of controllers provides satisfactory response for speed de-
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viation overshoot and transient oscillation with zero steady state error. Table III presents the closed-loop eigenvalues at normal load operating point and the same evaluated at heavy and light load operating point to verify the shift of dominant ones. Similar study is performed with gain matrix determined at light load operating point. Table IV presents the closed-loop eigenvalues determined at light load operating point. The observation suggests that location of closed-loop eigenvalues obtained from LQR method does not differ much with changes in operating points. It is also observed that with optimal PS approach; only dominant eigenvalues get shifted to new optimal location irrespective of any subsequent change in operating point. Thus, the control gain matrix determined using optimal PS approach is independent of system operating conditions, i.e., its variation do not adversely affect the controller performance. It is possible to select a single set controller gain matrix to ensure the stabilization of the system over a wide an operating range. Thus from the above discussion it is understood, optimal PS performance is as good as widely used LQR, with an advantage to ease in computation of control gain matrix. B. Estimator Based Controller Performance Linear optimal control and modern control theories introduced to improve the dynamic performance of systems however depend on the accuracy of the model. Its reliability gets reduced as the system becomes larger and complex. In use of state-feedback theories, the states are assumed to be measurable or are estimated. The following discuss the estimation of the states of SMIB system with noisy measurements. The process noise and measurement noise are considered to be uncorrelated and unbiased. The state (12)–(19) are solved using a fourth-order Runge–Kutta method with 64 steps taken between each observation. The closed-loop initial response of the compensated SMIB system with LQR and PS approach is shown in Fig. 10. The estimated states and state-error are shown in these figures. The observation is Fig. 10(a) suggests that the initial response of estimated states using LQR and PS control approach yield similar variation. Fig. 10(b) also illustrates the same finding. Further the comparison between the estimation techniques as shown in Fig. 10(a)–(b), it is understood that the state-error (error LQR and error PS) due to linear KF results in comparatively higher overshoot/undershoot, but with faster settling time. The KF is unable to completely estimate the state due to time delay problems and unmodeled dynamic problems and thus tends to over compensate. The comparison of control effort required in each estimation techniques is presented in Fig. 11. The control signals in case of states estimated by linear KF vary over a comparatively large magnitude but settles down quickly. This is a vivid example of the classical approach trade-off between response time and overshoot. In practice, a minimum control effort is desired to achieve sufficient damping of controlled signals. Furthermore, for quantitative analysis, the measure of state estimation techniques and optimal controller has been considered by defining the performance index as shown in the equation at the bottom of the next page, where defines the steadystate values following the initial condition test. The minimization of these values would provide improved dynamic perfor-
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Fig. 10. Estimated states and state-error variation with optimal control approach at heavy load operating condition.
Fig. 11. Control signal variation with optimal control approach at heavy load operating condition.
mance system and the computed values are shown in Fig. 12. The compensation by both PS and LQR using FDDF technique (i.e., PS-FDDF, LQR-FDDF) presents superior states estimation. This is concluded from the bar-chart representation of performance index as shown in Fig. 12(a). The quantitative analysis for required control effort to support the FDDF estimation technique is illustrated in Fig. 12(b). At normal/light load conditions, the computed control signal (PS-FDDF & LQR-FDDF))
becomes comparatively too small to be represented as a barchart. VII. CONCLUSION The hydro power plant control as SMIB system was studied in the paper. An approach towards achieving an optimal control using pole shift satisfying a quadratic performance index was
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Fig. 12. Bar-chart representation of performance index for state estimation techniques and optimal controller.
discussed. In this study, the damping of load angle and speed deviation via LQR and PS, when applied independently and also through state estimator was investigated and discussed for different operating conditions. The time response and frequency response results illustrated that the said method was effective to damp out the load angle and speed oscillations. It suggested a response as equivalent to linear quadratic regulator and in some cases better with an advantage to ease in computation of controller gain matrix. The simultaneous stabilization of the SMIB system was demonstrated by considering three different operating conditions as heavy load, normal load, and light load. The controller gain matrix determined at any operating point ensured shift of dominant open-loop eigenvalues at other operating points too. Thus, a control scheme provided great damping characteristics and enhanced significantly the system stability with even changes in plant operating conditions. Briefly, the system remained robust over a wide range of change in operating conditions. The state-feedback control approach is further illustrated with estimation of states using linear Kalman filter and first-order divided difference filter. The state-error based on linear Kalman filter estimation results in comparatively higher overshoot/undershoot, but with faster settling time. The comparison between the estimation techniques suggested that the FDDF provides a comparatively lower control effort but at expense of higher oscillation before settling down.
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Nand Kishor (SM’04–AM’06–M’07) received the Ph.D. degree from the Indian Institute of Technology, Roorkee, in 2006. He is an Assistant Professor in the Department of Electrical Engineering, Motilal Nehru National Institute of Technology, Allahabad, India. His main research includes power plant modeling and control, application of artificial intelligence in system modeling, and renewable energy systems. He has published his research in reputed international referred journals and proceedings and is a regular reviewer of manuscripts from referred journals of well-known publishers.
