1. Field of the Invention
The present invention relates generally to interconnected power generation systems, and more specifically, to a particle swarm optimizing sliding mode controller having a control optimization feature that reduces chattering in the responses of the interconnected power generation systems.
2. Description of the Related Art
The Automatic Generation Control (AGC) problem has been one of the most important issues in the operation and design of contemporary electric power systems. This importance is due to the role of the AGC in securing a satisfactory operation of power systems and ensuring constancy of speed for induction and synchronous motors, thereby improving the performance of generating units. The purpose of AGC is to track load variation while maintaining system frequency and tie line power interchanges (for interconnected areas) close to specified values. In this way, transient errors in frequency and tie line power should be minimized and steady error should not appear.
In the last two decades, many techniques were proposed for the supplementary control of AGC systems. Conventionally, Proportional Integral (PI) and Proportional Integral Derivative (PID) controllers are used for AGC. However, PI has many drawbacks, some of which are long settling time and relatively large overshoots in the transient frequency deviations. Furthermore, utilization of optimal control theory has been examined. The controller design is normally based on the parameters of the linear incremental model of the power system, which, in turn, depend on the condition of the power system. Therefore, the linear optimal controller is sensitive to variations in the plant parameters or operating conditions of the power system. Moreover, the linear optimal controller yields unsatisfactory dynamic response in the presence of Generation Rate Constraint (GRC). Other techniques used for designing the secondary control loop for the AGC include Neural Network methods, Superconducting Magnetic Energy Storage (SMES) unit applications and spline techniques. Furthermore, the application of the Sliding Mode Controller (SMC) to the AGC problem has already been considered in the art. SMC possesses some attractive features, mainly robustness and good transient response. As is known by those having ordinary skill in the art, a SM controller has been compared with conventional and optimal control methods for two equal area non-reheat and reheat thermal systems.
However, a systematic method for obtaining the switching vector and optimum feedback gain of the SMC has yet to be considered. Pole placement technique has been utilized in designing the SMC for a single area non-reheat AGC system in known prior experimentation in which the feedback gains were selected by trial and error.
In practice, AGC models are nonlinear. Unfortunately, conventional control design methods are not efficient when nonlinearities are introduced to the incremental models of control systems. Thus, other methods should be utilized for the design of the controllers. One of the most reliable techniques is the use of iterative, heuristic optimization algorithms, which can be used to tune the controllers to obtain their optimum settings. Some of the recent attempts that utilize the iterative heuristic algorithms for solving the AGC problems (for linearized models) can be found in previous work of the present inventor, where a Genetic Algorithm (GA) was used to optimize only the feedback gains of the SMC applied to a single area, non-reheat AGC. On the other hand, Particle Swarm Optimization (PSO) has used for the same purpose, and for a single area linear model, too, but there is still a need to have a systematic method for obtaining the switching vector and optimizing feedback gain of the SMC when using PSO.
Thus, a particle swarm optimizing sliding mode controller solving the aforementioned problems is desired.
The particle swarm optimizing sliding mode controller provides a sliding mode controller (SMC) with a chattering reduction feature applied to an interconnected Automatic Generation Control (AGC) model. The controller uses a method that formulates the design of SMC as an optimization problem and utilizes a Particle Swarm Optimization (PSO) algorithm to find the optimal feedback gains and switching vector values of the controller. Two performance functions are used in the optimization process to demonstrate the system dynamical performance and SMC chattering reduction. The tested two-area interconnected AGC model incorporates nonlinearities in terms of Generation Rate Constraint (GRC) and a limiter on the integral control value. Comparison with a previous AGC methods reported in literature validates the significance of the SMC design.
These and other features of the present invention will become readily apparent upon further review of the following specification and drawings.
Similar reference characters denote corresponding features consistently throughout the attached drawings.
The particle swarm optimizing sliding mode controller provides a system for controlling two non-linearly interconnected power generation systems. As shown in
The tie line power is x5. On the other hand, x7, x8, and x9 are, respectively, the incremental changes in generator output (p.u. MW), governor valve position (p.u. MW), and integral control of area 2. The control objective in the AGC problem is to keep the change in frequencies (Hz) Δf1=x1 and Δf2=x2, as well as the change in tie-line power ΔPtie=x9, as close to zero as possible when the system is subjected to load disturbance ΔPL1 and ΔPL2 by manipulating the inputs u1 and u2, respectively. The values of the parameters used for the interconnected system are as in Table 1.
The fundamental theory of the SMC is well known to persons having ordinary skill in the art. Different control goals, such as stabilization, tracking, and regulation, can be achieved using SMC by the proper design of the sliding surface. The regulation problem is addressed wherein the objective is to keep specified states as close to zero as possible. A block diagram of the SMC 200 for the regulation problem is shown in
{dot over (x)}(t)=AX(t)+BU(t) (1)
where:
X(t) is an n-dimensional state vector;
U(t) is an m-dimensional control force vector;
A is an n×n system matrix, and
B is an n×m input matrix.
The SMC control laws for the system of (1) are given by:
where the feedback gains are given as:
where ci's are the switching vectors that are selected by pole placement or linear optimal control theory.
The design procedure for selecting the constant switching vectors ci using pole placement includes a procedure that defines the coordinate transformation:
Y=MX (4)
such that:
where M is a nonsingular m×n matrix and B2 is a nonsingular m×m matrix. Then, from (4) and (5), calculating:
where A11, A12, A21, A22 are respectively (n−m)×(n−m), (n−m)×m, m×(n−m) and (m×m) submatrices. The first equation of (7), together with (3), specifies the motion of the system in the sliding mode, which is:
{dot over (Y)}
1
=A
11
Y
1
+A
12
Y
2 (8)
Σ(Y)=C11Y1+C12Y2 (9)
where c11 and c12 are m×(n−m) and (m×m) matrices, respectively satisfying the relation:
[c11c12]=cTM−1 (10)
Equations (8) and (9) uniquely determine the dynamics in the sliding mode over the intersection of the switching hyper-planes.
The subsystem described by (8) may be regarded as an open loop control system with state vector Y1 and control vector Y2 being determined by (9), that is:
Y
2
=C
12
−1
C
11
Y
1 (11)
Consequently, the problem of designing a system with desirable properties in the sliding mode can be regarded as a linear feedback design problem. Therefore, it can be assumed, without loss of generality, that c12=an identity matrix of proper dimension. Next, equations (8) and (11) can be combined to obtain:
{dot over (Y)}
1
=[A
11
−A
12
C
11
]Y
1 (12)
It is well know in the art that if the pair (A, B) is controllable, then the pair (A11, A12) is also controllable. If the pair (A11, A12) is controllable, then the eigenvalues of the matrix [A11−A12C11] in the sliding mode can be placed arbitrarily by suitable choice of C11. The feedback gains αij are usually determined by simulating the control system and trying different values until satisfactory performance is obtained.
Particle Swarm Optimization (PSO) is an evolutionary computation technique inspired by social behavior and bird flocking or fish schooling. As shown in
For each particle x, the desired optimization fitness function, J, is evaluated in d variables.
Then the procedure compares particle fitness evaluation with xpbest, which is the particle with best local fitness value. If the current value is better than that of xpbest, then set xpbest equal to the current value, and Xpbest locations equal to the current locations in d-dimensional space.
Next, compare fitness evaluation with population overall previous best. If the current value is better than gbest, the global best fitness value, then reset xgbest to the current particle's array index and value.
Update the time counter t, inertia weight w, velocity v, and position of x according to the following equations:
where wmin and wmax are the maximum and minimum values of the inertia weight w, m is the maximum number of iterations, i is the number of the particles that goes from 1 to n, d is the dimension of the variables, and α is a uniformly distributed random number in (0,1). The particle velocity in the dth dimension is limited by some maximum value vdmax. This limit improves max the exploration of the problem space. In this study, vdmax is proposed as:
v
d
max
=kx
d
max (17)
where k is a small constant value chosen by the user, usually between 0.1-0.2 of xdmax.
Loop to the particle swarm simulation step until a criterion is met, such as a good fitness value or a maximum number of iterations (generations).
The nonlinear interconnected AGC model has nine state variables and two control inputs. The switching vector of the sliding mode controller is represented by:
and the feedback gains are given by:
This will result in a total of 36 parameters to be optimized using the PSO based approach. As shown in
This performance index puts emphasis on improving the dynamical behavior of the AGC system. This has been done by including the absolute error time of the internal parameters of the AGC system into the performance index. Furthermore, to reduce chattering from the control signal, the controller provides a second performance index J2, which includes the deviation of the control effort:
Continuing with the procedure of
The inventors have compared the particle swarm optimizing sliding mode controller with a Linear Quadratic Regulator (LQR)-based system for two area interconnected nonlinear power systems. The sliding mode controller uses PSO for the selection of the optimum gains of this integral controller. The optimum gains are K1=0.5512 and K2=0.01. The PSO settings used are: number of particles=15; maximum number of iterations=500; wmax=0.9; wmin=0.4; and the maximum velocity constant factor k=0.1.
On the other hand, for the SMC, the optimum switching vector values and feedback gains were found to be as follows:
i) For objective function J1:
ii) For objective function J2:
After getting the above SMC parameters, the system is simulated for a 0.01 p.u. load disturbance in area 1. The fast convergence of the objective functions J1 and J2 are shown in plot 300 of
The frequency deviation in areas 1 and 2, as well as the tie-line power change of the interconnected system as compared to prior art methodologies (the LQR model) and the present optimal integral control, are shown in plots 400, 500, and 600 of
It is quite clear that using J as a performance function allows the design of a SMC controller with improved dynamic behavior, as seen in FIGS. 4-6. This is at the expense of increased fluctuations (chattering) in the control effort u1 plot 700 and control effort u2 plot 705, as seen in
In order to test the robustness of the controller, the model parameters Tp1, Tp2, Tg1, Tg2, Kp1, and Kp2 were changed by ±25%. The controller feedback gains and switching vector values obtained before when taking J1 as an objective function are kept the same. The dynamical behavior of the controller under the 0.01 p.u. load disturbance in area 1 is shown in plots 900, 1000, and 1100 of
An efficient sliding mode controller with a chattering reduction feature has been disclosed herein. The optimal feedback gains and switching vector values of the controller are selected based on formulating the SMC design procedure as an optimization problem. A particle swarm optimization algorithm is used as an optimization tool.
The controller has been applied effectively on a practical two-area AGC problem with nonlinearities in the model. The dynamical performance of the system when using the controller is highly enhanced as compared to the dynamics when using previous controllers. The chattering in the control efforts have been successfully eliminated using a new performance function. In addition, the controller has been found to be robust against parameter variations of the model.
It is to be understood that the present invention is not limited to the embodiment described above, but encompasses any and all embodiments within the scope of the following claims.