Adaptive And Self Tuning Power System Biology Essay

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Abstract- This paper is concerned with the investigation of an adaptive power system controller for a synchronous generator connected to an infinite bus through a double-circuit transmission line. Auto tuning stabilizers are required for effective control over a wide range of operating conditions. This can be done using self- tuning regulators (STR) which identify the system under different operating conditions and provide the control action accordingly. Simulation of a synchronous generator subject to major disturbances at different operating conditions, including three-phase short circuit, has been developed to demonstrate the effectiveness of the proposed controller. A comparative study has been effectuated between a fixed-gain PID power system stabilizer and the proposed adaptive stabilizers, for various changes in the operating conditions. It was shown that the adaptive PSS can stabilize the synchronous generator for different operating conditions, and it can damp the system oscillations in a short period.

Key words: Synchronous generator, Power system stabilizers, System identification, Model reference adaptive control, Self-tuning adaptive control.

Introduction

Voltage regulators were first designed to keep the voltage at the desired value with varying loads. It was noticed that in some cases the use of voltage regulators led to transient instability and machines had a tendency to oscillate continuously under certain loading conditions. Thus increasing attention has been focused on the effects of excitation control on damping of oscillations. In particular it has been found useful and practical to incorporate stabilizing signals derived from speed, frequency or power, superimposed on the voltage error signal to provide damping. A detailed analysis of damping and synchronizing torques of the synchronous machines with speed as a stabilizing signal is given in [Jagannathan Kanniah, O.P. Malik, G.S. Hope (part I)-1984],[ Jagannathan Kanniah, O.P. Malik, G.S. Hope (part II)-1984], [Adel A. Ghandakly, Ahmed M. Farhoud-1992], [Adel A. Ghandakly, Jiang J. Dai-1992],[ B.A. Archer, L.E. Midford, J.B. Davis-2002] and [A. Khodabakhshian-2003]. Digital simulation of a synchronous machine subject to a major disturbance is performed to demonstrate the effectiveness of the proposed adaptive controllers.

The conventional fixed-gain power system stabilizers are designed for one particular operating condition around which a linearized model is obtained [Yuan-Yih Hsu, Kan-Lee Liou-1986]. Because of the non-linear characteristic properties of the power systems, it is desired to design the stabilizer parameters to be automatically adjusted according to the system's operating conditions. Adaptive control theory offers technique for the design of such a device [Adel A. Ghandakly, Jiang J. Dai-1992].

This paper is concerned with the development of three different types of controllers for synchronous generator, a fixed-gain PID controller, a self-tuning pole-placement power system stabilizer and a model reference adaptive excitation controller. The system considered is a synchronous generator connected to an infinite bus through a double circuit transmission line. The adaptive control algorithms track the system operating conditions using a least-squares identification technique and the control input is calculated to satisfy the desired specifications [A. Khodabakhshian, K. Jamshidi-2003]. Digital simulation of a synchronous machine subject to a major disturbance at various operating conditions, and at a three phase short circuit is performed to demonstrate the effectiveness of the proposed adaptive controllers. Section (2) presents the design of a self-tuning pole-placement controller. Section (3) presents the design of model reference adaptive controller. System Identifications using recursive least square (RLS) is shown in section (4). Section (5) represents the design of a fixed gain PID controller. Section (6) represents the simulation studies, and the conclusions are presented in section (7).

Design of a Self-Tuning Pole-Placement Controller

The power system under investigation is a synchronous generator connected to an infinite bus through a double circuit transmission line.

Consider the system described by its input-output relationship

(1)

Where

A(z)= z5+a1 z4+ a2 z3+ a3 z2+a4 z +a5 (2)

B(z)= b0 z4+ b1 z3+ b2 z2+b3 z +b4 (3)

And a1, a2, a3, a4, a5, b0 , b1, b2, b3 and b4 are coefficients which can be estimated using the recursive-least-square (RLS) identification [A. Khodabakhshian, K. Jamshidi-2003] and [ P.E. Wellsted, M.B. Zarrop-1991]. The desired response is selected in order to give a satisfactory transient and dynamic performance. Then the desired closed-loop pulse transfer function is given

(4)

Note that the disturbance models are not used in the pole-placement design method. The disturbances are instead considered indirectly by introducing constrains on the model (Hm), the observer polynomial Ao(z), and the control law. Power system stabilizer has one output (U) and two inputs, the command signal (Uc) and the measured output (Y), a general linear structure for a regulator with these inputs and outputs [K.J. Astrom, B. Wittenmark-1997] may be represented by the following control law:

R(z). U(k) = T(z). Uc(k) - S(z). Y(k) (5)

The design of a power system stabilizer is thus reduced to the problem of finding the polynomials R, S, and T by solving the diophantine equation, that satisfy an additional requirements on the admissible controls. In order to obtain a self-tuning controller structure let's assume

R(z)=(z-1)(z+r1) (6)

S(z)=so z5+s1 z4+ s2 z3+ s3 z2+s4 z +s5 (7)

(8)

Ao = ao z2+a1 z+a2 (9)

Then the closed-loop characteristic equation becomes

R(z). A(z)+B(z). S(z)= Ao(z). Am(z) (10)

From equation(1) B = B+. B- , where B+ is a polynomial that contain all zeros inside the unit circle, and B- is a polynomial that contain all zeros outside the unit circle. The above procedure is repeated at each sampling instant where the system parameters are identified using a recursive-least-square algorithm.

Design of Model Reference Adaptive Controller

The design of a model reference adaptive PSS using input-output models is based on error system method. Due to the variation of the system parameters with the operating point, the parameter variations must be adapted using one of the adaptive algorithms. The control system is designed by assuming that the parameters are time invariant known values. The input-output relation of the system is:

A(z-1)Y(k) = B(z-1) U(k) + C(z-1) D(k) (11)

Where

A(z-1)= 1+a1 z-1+ a2 z-2+ a3 z-3+a4 z-4 +a5 z-5 (12)

B(z-1)= b0 + b1 z-1+ b2 z-2+b3 z-3 +b4 z-4 (13)

C(z-1)= c0 + c1 z-1+ c2 z-2+c3 z-3 +c4 z-4 (14)

Taking the first difference of equation (11)

A(z-1) ΔY(k) = B(z-1) ΔU(k) (15)

Where the first difference is represented by Δ= 1- z-1

Y(k) = Y(k-1) - a1 ΔY(k-1) - a2 ΔY(k-2) - a3 ΔY(k-3) - a4 ΔY(k-4) - a5 ΔY(k-5) + bo ΔU(k) + b1 ΔU(k-1) + b2 ΔU(k-2) + b3 ΔU(k-3) + b4 ΔU(k-4) (16)

Here Ym(k) is assumed to be a bounded output variable of a reference model, which is the desired signal for the controlled object. The following error system is derived from equation (16).

e(k) = Ym(k) - Y(k) (17)

e(k) = Ym(k) - Y(k-1) + a1 ΔY(k-1) + a2 ΔY(k-2) + a3 ΔY(k-3) + a4 ΔY(k-4) + a5 ΔY(k-5) - bo ΔU(k) - b1 ΔU(k-1) - b2 ΔU(k-2) - b3 ΔU(k-3) - b4 ΔU(k-4)

The control algorithm is obtained so that the error e(k+1) can be zero.

U(k)= [Ym(k) - Y(k-1) + a1 ΔY(k-1) + a2 ΔY(k-2) + a3 ΔY(k-3) + a4 ΔY(k-4) + a5 ΔY(k-5) - b1 ΔU(k-1) - b2 ΔU(k-2) - b3 ΔU(k-3) - b4 ΔU(k-4) ] / bo + U(k-1) (18)

In practice, the system parameters vary with the operating conditions and should be estimated successively using the parameter adjustment algorithm RLS. Then the adaptive control input is obtained by using the estimated parameters:

U(k)= [Ym(k) - Y(k-1) + 1 ΔY(k-1) + ΔY(k-2) + ΔY(k-3) + ΔY(k-4) + ΔY(k-5) - ΔU(k-1) - ΔU(k-2) - ΔU(k-3) - ΔU(k-4) ] / bo + U(k-1) (19)

System Identifications Using Recursive Least Square

In self-tuning control the parameter estimation scheme should be iterative, allowing the estimated model of the system to be updated at each sample interval as new data become available.

In particular, it is useful to be able to visualize the estimation process as shown in figure(1). In this scheme new input/output data becomes available at each sample interval. The model based on the past information is used to obtain an estimate of the current output. This is then compared with the observed output y(t) to generate an error ε(t). This in turn generates an update to the model which corrects to the new value . This recursive "predictor corrector" form allows significant saving in computation. Instead of recalculating the least squares estimate in its entirety, requiring the storage of all previous data, it is both efficient to store the old estimate calculated at time t, denoted by , and to obtain the new estimates by an updating step involving the new observation only. To see how this is done, compare a least squares estimate based on data from time samples 1 to t with the estimate based on data from time samples 1 to (t+1).

Figure(1) Adaptive Identification scheme

For the estimator using data from time 1 to t, it will be:

(20)

Write X(t) as function of time to indicate that it is based upon data from time steps to t.

The estimate at step (t+1) are then given by:

(21)

Introducing some shorthand by denoting

P(t)= (22)

B(t)= (23)

And then, (24)

(25)

Also,

(26)

And, (27)

Equation (27) gives a direct update from B(t) to B(t+1). The crucial step is to establish the same direct update from P(t) to P(t+1). The standard way to do this is by applying the Matrix Inversion Lemma:

From equation (26) assigning

, C =1, B = X(t+1), D = XT(t+1)

Give:

(28)

Equation (28) gives us the means to update P(t) to P(t+1) without inverting a matrix. In fact, the only inversion is of the scalar term

.

The recursion for P(t+1) can be combined with the recursion for B(t+1) equation (27) in many ways to give a direct recursion for from . The most common way is to define the error variable ε(t+1) as indicated in figure (page 63) by:

(29)

And substitute for y(t+1) in equation (27), this gives:

Substituting for B(t), B(t+1) using equations(24,25) gives

(t+1) (30)

The simulation results for the synchronous machine with voltage regulator and exciter as shown in figure(2).

Figure(2) Linearized incremental model of a synchronous machine with an exciter and stabilizer

Using the recursive least-squares identification technique to identify the system parameters, for various system's disturbances occurred on the synchronous machine (i.e., change in the operating point, short circuit,…etc). The identified parameters are shown in figure(3), according to the table(1).

Figure(3) Identified Parameters Equ.(16).

Design of a Fixed Gain PID Controller

The state equation of the system under a particular loading condition can be written as:

(31)

(32)

Where, X(t) = [Δδ(t) , Δω(t) , Δ , Δ , Δ]T is the state vector and Y(t) = Δω(t) is the output signal, for the system whose block diagram is shown in figure(2).

The control signal U can be expressed as:

Y(s)

(33)

Where is the washout time constant and , , and are the gains of the PID controller. There unknown parameter , , and can be determine as follows. From equations (31) and (32) obtain

(34)

(35) Combining equations (33),(34), and (35), to get

(36)

The eigenvalues of the closed-loop system equipped with PID controller are then the solutions of the

=0 (37)

Therefore, a set of four simultaneous algebraic equations with four unknown variables , , and can be obtained by substituting four pre-specified eigenvalues and into equation (37). The desired controller parameters , , and are then computed by solving these four algebraic equations. Usually, the pre-specified eigenvalues are obtained by shifting the four badly-damped eigenvalues leftward. As a result, the damping of the system could be improved through this eigenvalues shifting process since the eigenvalues of the closed-loop system have been shifted to more desirable pre-specified locations.

The eigenvalues of the closed-loop system equipped with such a fixed-gain PID controller are functions of the six constants which, in turn are functions of generator loading conditions. Hence these eigenvalues would deviate from the pre-specified values as long as the loading condition is different from the nominal condition. To show this phenomenon clearly figure (4) show the system pole locations with the loading conditions. The movement of the system dominant roots are according to the change in the loading conditions, are shown in figure (4).

Figure(4) Pole Shifting With Change in the Operating Point.

Simulation Studies

Consider the single machine infinite bus system. The most commonly used linearized model [P.M. Anderson, A.A. Fouad-1977] with voltage regulator and exciter included is shown in figure(2). The system desired response is selected to give a damping factor (0.3) and un-damped natural frequency (1.265 rad/sec), then the system dominant roots are (-0.4±j1.2), note that the characteristic equation of the closed-loop system is a 5th order, then construct the desired characteristic equation for the system and solving the resultant simultaneous algebraic equations to get the desired pole locations. Many case studies were performed to show the effect of changing the system conditions on the controller performance. Note that the eigenvalues of the closed-loop system are functions of generator loading conditions. Hence, these eigenvalues would deviate from the pres-specified values as long as the loading condition is different from the nominal condition, eigenvalue study is carried to show the effect of the loading conditions on the system eigenvalues as shown in figure(4), which show that at certain loading condition the system become unstable. A sample of the simulation results applying the above three controller techniques is presented as shown in figure(5)-figure(7). Figure(5), Figure(6), and Figure(7) shows the angular speed error using PID, self-tuning, and adaptive model reference power system stabilizer, when the system running under nominal operating conditions (a) (active power (P)=1.0 p.u, Reactive power (Q)=0.62 p.u. and a power factor (pf)= 0.85 lagging), (b) shows the angular speed error when the operating conditions are changed to ( P=0.5 p.u., Q= - 0.24 p.u. and pf=0.9 leading), (c) shows the angular speed error when the operating conditions are changed to (P = 1.1 p.u., Q= - 0.73 p.u., and pf= 0.9 leading). The system response in case of a three-phase fault at the midpoint of one of the two transmission lines (P = 1 p.u., Q = 0.62) p.u.) are shown in (d), and (e) shows the angular speed error response with pre-fault and post -fault conditions.

(a) (b)

(a)Normal Operating Condition P=1pu, Q=0.62pu.

(b) Deviation From Normal Cond. to P=0.5pu , Q= -0.24pu.

(c) (d)

(c) Deviation From Normal Cond. to P=1.1pu, Q= -0.73pu.

(d) Deviation From Normal Cond. to Three-Phase Short Circuit (P=1pu.).

(e) System Response With Pre-Fault and Post -Fault Conditions

Figure(5) The Change in Angular Speed With Fixed Gain PID Power System Stabilizer.

(a) (b)

(a)Normal Operating Condition P=1pu, Q=0.62pu.

(b) Deviation From Normal Cond. to P=0.5pu, Q= -0.24pu.

(c) (d)

(c) Deviation From Normal Cond. to P=1.1pu, Q= -0.73pu.

(d) Deviation From Normal Cond. to Three-Phase Short Circuit (P=1pu)..

(e) System Response With Pre-Fault and Post -Fault Conditions

Figure(6) The Change in Angular Speed With Self-Tuning Power System Stabilizer.

(a) (b)

(a)Normal Operating Condition P=1pu , Q= 0.62pu.

(b) Deviation From Normal Cond. to P=0.5pu, Q= -0.24pu.

(c) (d)

(c) Deviation From Normal Cond. to P=1.1pu, Q= -0.73pu.

(d) Deviation From Normal Cond. to Three-Phase Short Circuit (P=1pu)..

(e) System Response With Pre-Fault and Post -Fault Conditions

Figure(7) The Change in Angular Speed With Model Reference Adaptive Power System Stabilizer.

Conclusions

Conventional fixed-gain power system stabilizer works reasonably well over a medium range of operating conditions. However, the damping may diminish as the generator load changes or the network configuration is altered by faults which lead to deterioration in the stabilizer performance.

A Self-tuning pole-placement and model reference power system stabilizers for the excitation control of a synchronous generator have been presented in this paper. Results from the digital simulation show that adaptive controller can yield good damping characteristic over a wide range of operating conditions while fixed-gain PID controller is sensitive to the variation in the loading conditions. Improvement in transient stability limits can be also achieved by the proposed controllers.

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