Procedia Environmental Sciences 10 ( 2011 ) 350 – 355
doi: 10.1016/j.proenv.2011.09.057
Conference Title
Multi-machine Power System Robust Excitation Control
Jun Zhang, Xihuai Wang and Jianmei Xiao
Shanghai Maritime University, Shanghai, 200135, China
Abstract
Nowadays PID regulator, power system stabilizer (PSS) and nonlinear control were widely used in multi-machine power system excitation control. This paper will use the nonlinear robust control theory to design the excitation controller, in order to realize the decentralized robust control and improve stability of the system. Simulation results show that the nonlinear robust excitation controller is very effective for the improvement of system stability and disturbance attenuation compared with the PSS and the traditional PID controller.
Keywords: Multi-machine power system; Excitation control; Robust control; PID; PSS
1. Introduction
Recent years, with power systems increasingly operated closer to their transfer power limits, the dynamic response of stressed power systems under critical contingencies may result in complex nonlinear dynamic phenomenon, several modes of inter area oscillation have been identified. In addition, operating conditions of modern large-scale power systems are continuously varying in order to satisfy different load demands. As a result, control systems are required to enhance the system’s transient stability. The transient stability is concerned with a power system ability to reach an acceptable steady-state behaviour while loading demand change or after a major contingency occurs. The power system stabilizer (PSS) is usually regarded as an effective means to damp out the oscillations while improve the dynamic stability of the power systems.
Compared to the linear approximation method, nonlinear control theory applied to power system. In order to implement this type of controllers high-order total output derivatives have to be available and, even if all the states are available, they are typically corrupted by noise. A way to cope with these problems and simultaneously avoiding increasing the complexity of the machinery via sensors is combining the controller with a robust differentiator.
2011 3rd International Conference on Environmental
Science and Information Application Technology (ESIAT 2011)
© 2011 Published by Elsevier Ltd.
Selection and/or peer-review under responsibility of Conference ESIAT2011 Organization Committee.Open access under CC BY-NC-ND license.
1878-0296 © 2011 Published by Elsevier Ltd.
Selection and/or peer-review under responsibility of Conference ESIAT2011 Organization Committee.
2. Model of multi-machine power system
Consider a multi-machine power system with n+1 generators, where the ith generator is interconnected with the other n generators.
Under some standard assumptions, the dynamics of n interconnected generators through a transmission network can be described by classical model with flux decay dynamics [13]. The network has been reduced to internal bus representation assuming loads to be constant impedances and considering the presence of transfer conductance. The dynamical model of the i-th machine is represented by the classical third order model:
° ° ° ¯ ° ° ° ® ) ) ( ( 1 ) ) ( ( 2 , ' ' di di di qi fm di qi q qi s i i mi ii s i s i I X X E E T E I E D P H Z Z Z Z Z Z G (1) Where )} sin( ) cos( { E G I , 1 ' ' ii qi ij j i ij j i n i j j q qi
¦
E i G G G B G G z (2) )} sin( ) sin( { E B I , 1 ' ' ii di ij j i ij j i n i j j q qi Ei G G G B G G¦
z (3) Where G is the operating angle of the rotor, i Iqi and Idi represent currents in d–q reference frame of the i-th generator,i q
E is the transient EMF in the axis,
i f
E is the equivalent EMF in the excitation
coil, i d X and ' i d
X are direct axis reactance and direct axis transient reactance, respectively; Pmi is the mechanical input power assumed to be constant, Di is the damping factor; all parameters are in p.u. Hii,
represents the inertia constant, in seconds; i d
T is the direct axis transient short circuit time constant, in seconds i(t) is the rotor angle, in radians; Z represents the relative speed, i Zs 2 fs is the synchronous
machine speed, in rad/s; Gij and Bij are the i-th row and j-th column element of the nodal conductance
matrix and nodal susceptance matrix respectively, which are symmetric, at the internal nodes after eliminating all physical buses in p.u.. We consider
i f
E as the input of the system.
3. Robust Control Theory and Design Methods
H control theory was developed in recent years, it can inhibit the disturbance of linear system on the
system output. Following firstly briefly introduce the principles and procedures of using the method of Riccati based on H norm to design H robust control.
For the controlled system in the equation (1), selecting the output equation:
» » ¼ º « « ¬ ª u R x Q Z 5 . 0 5 . 0 (4)
In practice, an accurate model of power system is not available and a third-order model could not represent the generator unit precisely. For this reason, it is required to investigate the robustness of the
proposed controller with system parameter variation and model errors. The variations of system parameters are considered for robustness evaluation of the proposed controller.
From the simulations it can be seen that it is not necessary to have an accurate model for designing the controller and furthermore the controller is not sensitive to model errors. A robustness test has been carried out by changing the inertia constant H and the time constant T do of all machines, from their original values. Comparing the system responses with those simulated under normal conditions, it can be seen that the proposed scheme can still provide consistent control performance even if system parameters have changed. These responses show that the control strategy performs well, due that the proposed
control depends only on knowledge of the constants C, Km and KM of the system (5), which are
associated to each subsystem (3) of the multi-machine power system. Any parameter variation do not affect the performance of the control law.
Q and R is respectively the weight matrix corresponding to LQR control theory with quadratic performance index. A complete control system is made up with the equation (2) and (3):
° ¯ ° ® » » ¼ º « « ¬ ª u R x Q Z G Bu Ax x 5 . 0 5 . 0 Z (5)
For the control system (5), assumed (A, B) is stableˈand (A, Q0.5)is observable; Then it can be
introduced the following state feedback:
u=-Fx (6) Where F is the feedback gain matrix. Then through the calculation, we can get:
Z=Hx (7) Where: » » ¼ º « « ¬ ª R F Q H 0.5 5 . 0
Thenˈclosed loop transfer function from w to z can be expressed as: G BF A sI H s T( ) ( )1 (8) Find the solution P of the following Riccati equation (13), then according to equation (14) get the feedback gain matrix F.
0 1 1 2 PA PGG PBRB P Q P AT T T T J (9) P B R F 1 T (10) H norm, based upon the Riccati method to prove that after the introduction of the new feedback system
is stable, and closed-loop transfer function of the H norm:
2 2 sup Z z T f (11) Satisfying the inequality:
J
Where 2 inf Z O x Qx u Ru T T f
To minimize disturbance to the adverse effects of the system, Ȗ should try to select a smaller value. 4. Design of decentralized robust excitation controller
The purpose of this paper is using the H control theory to design the excitation controller of the power
system of the multi-machine, which can realize the decentralized robust excitation control and improve the stability of the system.
The following will be shown in Figure 1 about 4-machine power system to illustrate that the design method of controller. The components parameters and operating status of the system has been detail introduced in the literature.
Fig 1. A typical wiring diagram of four-machine power system
Refer to the modelling method described in the above section, we can obtain the equation of excitation of the control system in the current state of each generator and component parameters of the disturbed conditions: i i G
Z
i i i i i Ax Bu x i=1, 2, 3, 4 (12)According to the desired control objectives, selecting appropriate generators for each output vector z and the gain Ȗ.
» » ¼ º « « ¬ ª i i i i i u R x Q Z 0.5 5 . 0 (13)
Here, according to the H control theory introduced in the section 2, obtained for the previous control
system with disturbances of the generator model design controller. By solving the corresponding Riccati equation generator can get the corresponding feedback gain matrix.
Next, in accordance with Section 2 describes the HĞ control theory, get in front of the generator for disturbed control system model design the controller. By solving the corresponding Riccati equation generator can get the corresponding feedback gain matrix.
5. Simulation
To test this role and effectiveness of control methods, we are cited in a 4-machine system is simulated. For comparison, in addition to the proposed controller, it also means designed in accordance with traditional PID controller and PSS excitation controller. Simulation results are shown as Fig2, to Fig.4.
Fig 2. Simulation of regular PID
Fig 3. Simulation with PSS
6. Summary
Compared with other simulation results (limited space, not all for example), such as static stability and load disturbance, we can draw the following conclusions: PSS and the REC is better than PID in improving the voltage regulation accuracy, improve the static power system, transient stability and Inhibit the transmission line changes. And compared with DFLEC, the controller structure of REC is simple and easy to implement with greater advantages. Simulation results show that the nonlinear robust excitation controller is very effective for the improvement of system stability and disturbance attenuation compared with the PSS and the traditional PID controller.
Acknowledgements
Financial research support from the Key Discipline Project of Shanghai Municipal Education Commission under grant J50602, and the Research Project of Shanghai Maritime University under grant 09-15 are greatly appreciated.
References
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