This paper presents a new robust technique for tuning the parameters of a state feedback controller for load frequency control (LFC) of interconnected power systems using sequential quadratic programming (SQP) method. In this method the frequency deviation of the system is directly utilized to tune the controller parameters. The simulation studies are carried out for two-area interconnected power system. The Comparative results of the proposed method and a conventional PI controller show its robustness with a satisfactory response when the parameters of the system change.
Keywords-load frequency control; sequential quadratic programming; robust state feedback controller.
Introduction
Load frequency control (LFC) is of importance in electric power system design and operation. The objective of the LFC in an interconnected power system is to maintain the frequency of each area and to keep tie-line power flows within some pre-specified tolerances by adjusting the MW outputs of the LFC generators so as to accommodate fluctuating load demands.
There have been increasing interests for designing load frequency controllers with better performance during past years and many control strategies have been developed for LFC. The first proposed control strategy was a proportional integrator (PI) controller, which is widely used in the industry nowadays [1, 2]. The main drawback of this controller is that the dynamic performance of the system is limited by its integral gain. A high gain may deteriorate the system performance causing large oscillations and instability. Thus, the integral gain must be set to a level that provides a compromise between a desirable transient recovery and low overshoot in the dynamic response of the overall system [3]. However, because of the inherent characteristics of the changing loads, the operating point of a power system changes continuously during a daily cycle. Thus, a fixed controller may no longer be suitable in all operating conditions. Therefore, a lot of approaches have been reported in the literature to tune the gain of the integral controller. An extended integral control was proposed in [4] to obtain zero steady-state error as well as having a controlled overshoot in system performance. Fuzzy PI controllers were suggested in [5] for load frequency control of power systems. Moon [6] observed that a differential feedback in LFC can indeed improve system damping, and [7] proposed a derivative structure that can achieve better noise-reduction than a conventional practical differentiator. In [8-10] the design and tuning of PID load frequency controllers were reported.
With the increase in size and complexity of modern power systems, the system oscillation might propagate into wide area resulting in a wide-area blackout. Therefore, advanced control methods were applied in LFC, e.g., optimal control [11, 12], variable structure control [13, 14], and intelligent control [15, 16].
Usually a linear model around a nominal operating point is used in the load frequency controller design. However, because of the inherent characteristics of changing loads, the operating point of a power system changes very much during a daily cycle. Therefore, a fixed controller which is optimal under one operating condition may not be suitable in another status unless some precautions are considered.
Therefore, robustness becomes a main issue in the attempt to design a controller to satisfy the basic requirements for zero steady state and acceptable transient frequency deviations. Many robust control methods have been applied to load frequency control problem, for example, H-infinity control [17], µ-synthesis approach[18], robust pole assignment approach [19].
In this paper, a new robust state feedback controller for load frequency control is proposed. This technique is based on Sequential Quadratic Programming (SQP) method [18] in which the nonlinearity and parameter changes can be easily handled. It directly utilizes the frequency deviation of the system to tune the controller parameters. Comparative results will be given for the classic and proposed controllers. The proposed controller is robust and its dynamic performance is considerably desirable when the parameters of the system change.
Power System Model
Interconnected power systems naturally consist of complex and multi-variable structures with many different control blocks. They are usually non-linear, time-variant and/or non-minimum phase systems. Power systems are divided into control areas connected by tie lines. In each control area, the generators are supposed to constitute a coherent group. It means that the movements of their rotors are closely related [20]. Experiments on the power systems show that tie-line power flow and frequency of the area are affected by the load changes at operating point. Therefore, it can be considered that each area needs its system frequency and tie-line power flow to be controlled [21]. Additionally, it is desired that transient frequency oscillations without a large increase in the magnitude and speed control be reduced [22].
Generally, the load-frequency control is accomplished by two different control actions in interconnected power systems: (a) the primary speed control and (b) supplementary or secondary speed control actions. The primary speed control performs the initial vulgar readjustment of the frequency. By its actions, the various generators in the control area track a load variation and share it in proportion to their capacities.
The supplementary speed control takes over the fine adjustment of the frequency by resetting the frequency error to zero through an integral action. The relationship between the speed and load can be adjusted by changing a load reference set-point input. In practice, the adjustment of the load reference set-point is accomplished by operating the speed changer motor. The output of each unit at a given system frequency can be varied only by changing its load reference, which in effect moves the speed-droop characteristic up and down.
This control is considerably slower and goes into action only when the primary speed control has done its job. Response time may be of the order of one minute. The speed-governing system is used to adjust the frequency. Governors adjust the turbine valve/gate to bring the frequency back to the nominal or scheduled value. Governor works satisfactorily when a generator is supplying an isolated load or when only one generator in a multi generator system is required to respond to the load changes. For power and load sharing among generators connected to the system, speed regulation or droop characteristics must be provided. The speed-droop or regulation characteristic may be obtained by adding a steady-state feedback loop around the integrator.
In state feedback control, to force the steady state of Δf(t) to tend to zero, the integral of area control error (ACE) is used as an additional state, it is defined as:
DE (t) = ò ACE dt = ò (DPTie + β.Df) dt (1)
Where β = D+1/R.
The schematic of two-area interconnected power system with considering governor limiters for the uncontrolled case is shown in Fig.1. The overall system can be modeled as a multi-variable system in the following form:
(2)
Two-area power system
Where, A, B and W are the system, input and disturbance matrices, respectively. The x(t), u(t) and ΔPL are state, control and load changes disturbance vectors, respectively and represented as:
ΔPref1 and ΔPref2 are output of state feedback controller that obtained from equation (3):
(3)
Where K is state feedback matrix.
Tuning of feedback matrix parameters is done in the optimization process and will be discussed in the next section.
Tuning Methodology
Tuning methodology proposed in this paper is based on the fact that the frequency in each area and tie-lines power following a disturbance should remain constant.
If the frequency and tie-line power are constant, according to equation (1) area control error (ACE) is also constant (zero). This is represented graphically by a straight line in Fig. 2. Realistically this would not be possible and frequency and tie-line power and therefore ACE, would usually oscillate after a disturbance (load change) like the actual curve shown in Fig. 2.
Ideal and actual frequency curve
The proposed technique in this paper is to utilize this graphical information directly and to minimize the difference (error) between the actual ACE curve and the ideal ACE curve. By minimizing the difference between the curves over a specified time period, a setting of the controller parameters can be obtained which ensures that the frequency and tie-line power are as close to the ideal as possible.
The objective function F to be minimized is largely based on the integral squared error (ISE) formulation and is sum algebraic values of the error between the actual and ideal ACE curves within the shaded regions indicated in Fig. 2 and is given as:
(4)
Where ACEan and ACEin are the actual and ideal ACE curves of nth area, respectively and wn is weighting factor of nth area. By increasing the weighting factor of a particular area, the optimization routine will pay more attention to optimizing the performance of that specific area. The result obtained would be therefore skewed towards the stabilization of certain area [23].
Optimization algorithm
Mathematically, the optimization problem can be written as:
(5)
Where the parameters 'x' to be optimized are state feedback controller gains given in equation (3), f(x) is the objective function shown in equation (4) and g(x) are inequality constraints. In this problem inequality constraints are bounds of state feedback controller gains and n is number of state feedback gains.
This problem can be solved by using the Sequential Quadratic Programming (SQP), which is discussed below.
Sequential Quadratic Programming
The Sequential Quadratic Programming (SQP) is an optimization method that can be applied to solve a multi objective problem with different constraints. The name SQP comes from the meaning that one Quadratic Programming (QP) is solved for every each major iteration. The SQP is also known as Iterative Quadratic Programming, Recursive Quadratic Programming, and Constrained Variable Metric methods.
This optimization methodology has the ability to transform the problem into sub-problem that can be solved, and used as the basis of an iterative process [24].
SQP Implementation
The SQP algorithm can be summarized as follows;
At iteration n = 1,
Step 1: With considering the state feedback controller gains in current iteration (xn) (see equation (3)) and the current approximate Hessian (Hn), the Quadratic Programming (QP) sub-problem is formed. The QP is defined as:
(6)
Where, d is the search direction at the current iteration. The matrix Hk is a positive definite approximation of the Hessian matrix of the Lagrangian function (). The Lagrangian of this problem is defined as;
(7)
Where, λ is vector of approximate Lagrange multipliers.
Step 2: Solve the QP sub-problem to obtain dn. The dn is the solution of equation (5) at the iteration n, and is used here to determine the next iterate (xn+1) as;
(8)
Where αn is the step length parameter, which is determined in order to produce a sufficient decrease of merit function.
The merit function is defined as;
(9)
Where, ri is known as the penalty parameter and is initially set to ∂f(x)/∂gi(x).
Step 3: Update the Hessian approximation Hk+1 using the Broyden, Fletcher, Goldfarb, and Shanno (BFGS) [25] formula as;
(10)
Where sk = Xk+1-Xk and
Step 4: Stop if the convergence criterion is satisfied, which means sn becomes zero or very near to zero. Otherwise, set n=n+1, go back to Step 1 and continue.
Simulation Result
Simulation studies were performed on a two-area power system that explained in section 2. Typical data for the system parameters and governor limiters can be given as follows [8]:
Area#1: H=5, D=0.6, Tg=0.5, Tt=0.2, R=0.05
Area#2: H=4, D=0.9, Tg=0.6, Tt=0.3, R=0.0625
and T12=2
=0.4 =1.5
=1.2 =0.4
For the conventional PI controller by using the method given in [2] it was found that KI1=KI2=0.3 were the best selections for having the best performance.
Using SQP method for LFC design, the following results were obtained for state feedback controller parameters;
To test the proposed method, a step load change of 0.01 p.u., (i.e. ΔPL=0.01) is applied to the system area 1. Two cases were considered as follows;
Case 1: The system parameters are the nominal parameters given above.
Case 2: This case is employed to investigate the effect of changing the system parameters on system performance. Two different system parameters are considered as follows;
a) 25% increase for all system parameters. (Upper bound)
b) 25% decrease for all system parameters. (Lower bound)
Frequency deviation of area 1, frequency deviation of area 2 and tie-line power variation in nominal condition following the load changes using the proposed controller are shown in Fig. 3, 4 and 5 respectively when compared with the PI-conventional controller.
Frequency deviation of area 1
Frequency deviation of area 2
Tie-line power deviation
It should be mentioned that although the overshoot of frequency response of classical method shown in Fig. 3 is better than the state feedback method, the settling time of the latter is better than the former. Generally, by looking at Fig. 3 to 5 it can be concluded that the proposed method gives a better performance than the classical LFC.
Fig. 6 and 7 show response system under condition (a) including frequency deviation of area 1 and tie-line power deviation, respectively.
Frequency deviation of area 2 for upper bound of parameters
Tie-line power deviation for upper bound of parameters
As shown in Fig. 6 and 7, the system response for the proposed method has much less overshoot and settling time than the classic method.
Fig. 8 and 9 also show responses of frequency deviation of area 1 and tie-line power deviation for condition (b), respectively in which it can be found that the proposed method works much better than the classical PI controller.
Frequency deviation of area 1 for lower bound of parameters
Tie-line power deviation for lower bound of parameters
Conclusion
In this paper a new robust state feedback controller for load frequency control in a multi-area power system using sequential quadratic programming method has been developed.
Simulations were done on a two-area power system with considering governor limiters. The robustness of the proposed method is tested against change of parameters and results are compared with classical PI controller. Results indicated that the proposed method has a very desirable dynamic performance even when the system parameters change.
