International Journal of Electrical, Electronics and Computer Systems (IJEECS)
________________________________________________________________________________________________
The Effect of The Optimal Pole Locations on The Controlled Automatic Voltage Regulator System
1Ahmed A. M. EL-Gaafary, 2Ashraf M. Abdelhamid
1Electrical Engineering Dept., Faculty of Engineering, Minia University Minia, Egypt
2Communications and Electronics Dept. College of Engineering, Umm Al-Qura University Makkah, K.S.A
Abstract— The Automatic Voltage Regulator (AVR) problem is considered, a digital controller is designed using pole placement technique, in order to force the system to the stability region and improve system stability and specifications. The system dynamic response is computed and plotted to show the effect of the controller in improving system stability and performance, also bode plot and safety margins computed to assess system robustness.
In this study two methods are used in order to select pole locations, the dominant second order poles method, and the ITAE-prototype poles method, in which method different values of natural frequency suggested, to study its effect upon the system robustness.
Keywords — pole placement; digital controller; automatic voltage regulator; optimal control.
I.
INTRODUCTION
The availability of digital controllers has permitted more expensive computation of power system stability. This case gives an application of digital control on the voltage control or excitation control systems. The basic function of an excitation system is to provide direct current to the synchronous machine field winding. In addition, the excitation system performs control and protective functions essential to the satisfactory performance of the power system by controlling the field voltage and thereby the field current. The excitation system should contribute to effective `control of voltage and enhancement of system stability. It should be capable of responding rapidly to a disturbance so as to enhance transient stability. Fig.1 shows the functional block diagram of a typical excitation control system for a synchronous generator [1], [2].
Fig. 1 Functional block diagram of a synchronous
generator excitation control system [1]
II.
PROPLEM FORMULATION
Fig.2 shows the representation of the system in block diagram form, each block represents the transfer function for each component of the system. This system known as type 1 excitation system.
Fig.2 Block diagram of an excitation system.
The state space equations for this continuous time system can be derived in matrix form as follows [1], [2], [5].
(2) A. Background of Pole Placement
In pole placement technique, instead of using controllers with fixed configuration in the forward path or feedback path, control is achieved by feeding back the state variables through real constant under the restriction that system is completely controllable [6].
Consider a linear dynamic system in state space form as:
If the system is completely controllable pole placement can be used for stabilizing the system or improving its transient response. Here the control signal u can be represented as a linear combination of the state variables, that is
where 𝑲 in Eqn.5 is the state feedback gain matrix and is equal to: [ 𝑲1 𝑲𝟐 𝑲𝟑 𝑲𝟒 ]. Now the closed-loop system in Eqn. 3 can be represented as [10]
B. Discrete State Space Model
Using MATLAB program to convert the system continuous state space equations to discrete state-space ones for a specific values of the system constants, in order to designing a digital controller to the continuous time system [3].
The computed discrete state space model in Eqns.7 and 8 is:
C. Checking System Stability
There are many methods to check system stability.
MATLAB program may be used to compute the closed loop system poles which obtained to be equal to
from the obtained closed loop poles it seems that there are two of them lie outside the unit circle, it means that the system is unstable.
Fig.3 Step response for the discrete-time system.
The system time response to input value Vref of 0.01 pu, which ensure the same result, see Fig.3.
III.
CONTROLLER DESIGN USING POLE
PLACEMENT TECHNIQUE
The schematic of a full-state feedback system is shown below in Fig.4.
Fig.4 Discrete time system representation with
controller.
Assume that all the states are measurable, and then the computation of feedback control matrix k required to force the system to the stability region and with satisfactory specifications.
A. Checking System Controllability
Controllability is a structural property of the system, which determines whether any control sequence will drive the system between the specific states.
The necessary and sufficient condition providing that the closed loop poles can be placed at any arbitrary location in the z-plane is that the system is completely state controllable. In this condition the rank of the test matrix for controllability [ B AB ... An-1B] must be equal to the number of unknowns [8], [9].
𝑥 𝑘 + 1 = 𝐴𝑥 𝑘 + 𝐵𝑢(𝑘) (7)
𝑦 𝑘 = 𝐶𝑥(𝑘) (8)
𝑍1,2= 1.0229 ± 𝑗0.0852 𝑍3,4= 0.7912 ± 𝑗0.066
𝑥 𝑡 = 𝑎 𝑥 𝑡 + 𝑏 𝑢(𝑡) (3) 𝑦 𝑡 = 𝑐 𝑥 𝑡 + 𝑑 𝑢(𝑡) (4)
𝑢 = −𝐾 𝑥(𝑡) (5)
𝑥 𝑡 = 𝑎 − 𝑏 𝑥 𝑡 (6)
MATLB can be used easily to ensure that this system is completely state controllable
B. Selection of Desired Pole Locations
Assuming that the maximum overshoot of the system required to be no more than 5% and a settling time of less than 2 sec, which leads to choose the value of the damping ratio (ξ) to be 0.7 and the natural frequency (ω-
n) of 5rad/sec, we can use two different methods to select pole locations, the first one is the dominant second order poles method and the second is the ITAE prototype method with a sampling time Ts of 0.01 sample/sec.
1) The dominant second order poles method a) Determining the desired closed loop system poles: The desired dominant poles are determined in s- plane and calculated in the corresponding z-plane to be equal to:
And assume that the two other poles are real and lie far to the left at ten times the value of the real part of the two dominant poles which can be determined in z-plane to be equal to
b) Calculating the controller gain k: Ackerman's formula can be used by the aid of MATLAB in order to determine the gain matrix k which obtained to be equal to
c) The controlled system step response: Obtain the controlled system step response to a reference input Vref of 0.01pu, and with a suitable value of a feed forward gain which shown in Fig.5.
Fig.5 Step response of the controlled system with method 1.
Fig.5 shows that the steady state value of the terminal voltage equal to the reference input (i.e., the steady state error is equal to zero).
d) Determining system specifications: System specifications can be calculated and summarized in
TABLE.1, these results show that the digital controller achieves the desired system specifications.
TABLE.1SPECIFICATIONS OF THE AVR CONTROL SYSTEM WITH METHOD 1.
tr ts tp Mp
0.66 sec 1.3sec 0.89sec 4.8%
e) Studying the robustness of the controlled system:
Checking the closed loop poles gives a binary assessment of stability. In practice, it is more useful to know how robust of fragile stability is. One indication of robustness is how much the loop gain can change before stability is lost.
Bothe, the gain margin and phase margin give an estimate of the safety margin for closed loop stability.
The smaller the stability margins, the more fragile stability is.
Study the robustness of this design, using margin function and obtain the phase and gain margins [4], [7].
The results are:
Fig.6 Bode plot of the controlled system with method 1 Also the bode plot of the controlled system obtained in Fig.6. From the results it seems that this design is robust and the controlled system can tolerate a 23.7 dB increase or a 152 deg without going unstable.
2) ITAE-prototype poles
In this method of selecting closed loop poles the numerically evaluated poles determined to minimize the integral of the time multiplied by the absolute value of the error (ITAE), the optimum characteristic equation of the fourth order system with ωn=5rad/sec are determined [7]
a) Controller design: The determined optimum characteristic equation is used to get the gain matrix required k by Ackerman's formula to be equal to:
𝑍3,4= 0.965 10= 0.7003
𝐾 = −0.8662 0.207 0.0266 0.2424
𝑚𝑎𝑟𝑔𝑖𝑛𝑠 = 𝐺𝑚 𝑊𝑐𝑔 𝑃𝑚 𝑊𝑐𝑝 (9) 𝑚𝑎𝑟𝑔𝑖𝑛𝑠 = 15.36 152.67 20.32 1.587
k = [ -2.61 0.1316 -0.0267 1.5 ]
b) System step response:The step response for the controlled system with reference input obtained in Fig.7, from which figure it is clear that the system is stable.
Fig.7 Step response of the controlled system with method 2
c) Determining system specifications: System specifications can be calculated and summarized in TABLE.2, these results show that the digital controller achieves the desired system specifications.
TABLE.2SPECIFICATIONS OF THE AVR CONTROL SYSTEM WITH METHOD 2.
tr ts tp Mp
0.89 sec 1.35sec 1.02sec 2.38%
d) Studying the robustness of the controlled system:
Study the robustness of this design, using margin function and obtain the phase and gain margins. The results are:
Also the bode plot of the controlled system obtained in Fig.8. From the results it seems that this design is robust and the controlled system can tolerate a 5.34 dB increase without going unstable.
Fig.8 Bode plot of the controlled system with ωn=5
rad/sec.
e) Studying the effect of changing the natural frequency on the controlled system: In this method of choosing pole locations, system specifications can be slightly improved if 7 rad/sec natural frequency is taken.
The gain matrix required k is determined to be:
And the system specifications calculated and mentioned in TABLE.3.
TABLE.3SPECIFICATIONS OF THE AVR CONTROL SYSTEM WITH METHOD 2 WITH WN=7RAD/SEC.
tr ts tp Mp
0.627 sec 1.03sec 0.7063sec 2.6%
Also the controlled system bode plot shown in Fig.9
Fig.9 Bode plot of the controlled system with ωn=7 rad/sec.
IV.
CONCLUSION
In this paper the Automatic Voltage regulator (AVR) control system is studied. A digital controller is designed by the pole placement method. The desired closed loop poles are chosen using two different methods, the dominant second order poles approach, in which approach the desired closed loop poles located depending on the desired natural frequency and damping ratio, the second one is the ITAE prototype approach, with different values of natural frequency in order to realize a desired specifications.
All these designs are realized an accepted specifications mentioned in Tables 1,2 and 3. The results show robust controllers design are obtained, these controllers stabilizes the voltage regulator with no steady state error.
Comparing the results of the two methods at the same natural frequency 5rad/sec , the first method leads to a less rising time, settling time and time to peak but on the other side a more overshoot obtained.
In order to get more improvements by ITAE procedure the natural frequency must be increased to 7 rad/sec. In 𝑚𝑎𝑟𝑔𝑖𝑛𝑠 = 1.85 Inf 6.088 NaN
k= [-2.2631 0.1271 -0.0224 1.1954]
this case system specifications improved to the most better specifications ( the smallest rise time, settling time, time to peak, and maximum overshoot ), while the stability remains the same as that for ωn=5 rad/sec.
It’s important to choose the natural frequency and check the system robustness because it's affected by the desired closed loop pole locations.
V.
APPENDIX
System parameters definitions and values KA Regulator gain 57.14
TA Regulator amplifier time constant 0.05 sec KE Exciter constant related to self-excited time
constant -.0445
TE Exciter time constant 0.5 sec KG Generator gain 1
KR Regulator input filter gain 1
TR Regulator input filter time constant 0.05 sec
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