Optimal Controller for Single Area Load Frequency Control via LQR and Legendre Wavelet Function

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Figure 2. LFC for Single Area Power Control System

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3. Linear Quadratic Optimal Problem

Optimal control is an important science that deals with nonlinear optimal control problem (OCP) and the main objective of optimal control is to find an optimal controller that can be applied to a nonlinear system as well as to a linear and time varying system to minimize a certain cost function within the system's physical constraints. In other words, the main task of an optimal controller is to transfer the system from its initial state to the final state with minimum cost function ^[11]. Generally, optimal control can be classified as direct and indirect methods. Indirect methods are based on converting OCP into two-point boundary value problem (TPBVP), then solving the last one by Euler Lagrange technique or Hamilton-Jacobi-Bellman equation (HJBE). Direct methods can be implemented using discretization or parameterization methods. Mathematically, the optimal control problem can be considered as finding the optimal controller that minimize a performance index such

(2)

Subject to linear constraints and initial condition

(3)

Where A,B are n×n and n×m constant matrices respectively. Q is an n×n positive semidefinite matrix and R is an m×m positive definite matrix.

3.1. Optimal Control via Legendre Wavelet Function

One of the most famous indirect optimal control techniques that are used recently to approximate linear time invariant (LTI), linear time varying (LTV) and nonlinear systems are wavelet functions ^[12]. Legendre wavelet function has been used to approximate different systems within the subspace of trained initial condition and is considered a powerful function to give accurate approximation. Legendre wavelet function is defined such as ^[13].

Where P_m is the Legendre polynomial of order m; n refers to the section number, n=1,2,…2^K-1 ; K is the scaling parameter and can assume any positive integer and t∈[0,1]. Legendre polynomial can be defined as in [16] as follows:

(5)

State and control variables can be approximated using Legendre scaling function as follows:

(6)

(7)

3.2. Quadratic Programming Problem

Here, the quadratic form of optimal problem is constructed and may be solved easily using Matlab ^[13].

(8)

Subject to equality constraints

(9)

where

The optimal vector z* can be also calculated by the standard quadratic programming method ^[13] as follows:

(10)

3.3. Optimal Control via LQR

The linear quadratic regulator (LQR) is used to design optimal controllers for linear systems with quadratic performance index. LQR can be realized by finding optimal closed-loop state feedback control. In this article, LQR is applied to determine an appropriate state feedback gain along with the minimum variance method to design an LFC for single area power system. LQR can achieve good performance with minimum cost and can be easily designed using MATLAB. On the other hand, LQR has many limitations such as: it requires full state feedback, its function is only to guarantee stability, and it uses quadratic indices. This paper proposes a Legendre wavelet function to design optimal controller and compares its performance with an LQR controller.

4. Simulation Results

A comparison between an optimal controller based on Legendre wavelet function and an LQR control is made with the purpose of controlling LFC. The comparison is made in terms of performance with respect to frequency deviation , settling time and steady state error . The parameters of the numerical example which is solved using Simulink/Matlab and programming subscripts are shown in Table 1. In addition, the solution of an example consists of three scenarios: the first one contains no controller, the second solution uses an LQR control, and the final scenario uses an optimal controller based on Legendre wavelet function.

Table 1. Parameters of single area power station [11]

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The state space system and performance index of above example are respectively as follow ^[11]:

(11)

(12)

4.1. First Scenario

In the first scenario, a block diagram of LFC is constructed using Simulink and solved without using any controller with the power increasing by 200 MW (0.2 p.u). Figure 3 and Figure 4 show the Simulink block diagram of LFC and the frequency deviation respectively; while Table 2 summaries the obtained results.

Figure 3. Simulink block diagram of LFC without controller

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Table 2. Results of first scenario for LFC

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Figure 4. Frequency deviation of first scenario for LFC without controller

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4.2. Second Scenario

In the second scenario, an LQR optimal controller is designed to approximate the state feedback gain vector. The LQR controller is designed using M-file and Simulink. The block diagram and the frequency deviation response are shown in Figures 5 and 6 respectively. Table 3 summarizes the obtained results. However, the optimal gain vector k=[6.41 1.10 -112.60] is obtained at end of simulation.

Figure 5. LQR block diagram

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Figure 6. Frequency deviation of 2nd scenario for LFC with LQR

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Table 3. Results of second scenario for LFC using LQR

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4.3. Third Scenario

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In the third scenario, an optimal controller based on Legendre wavelet function is obtained using the approximation techniques to design the feedback gain vector K by converting the problem into a quadratic form that can be easily solved using Matlab. Furthermore, approximation via Legendre function is based on two factors: the order of Legendre polynomial (M) and the positive integer scaling parameter (K) which increases the accuracy of approximation proportionally. In this paper, the problem is solved for M=4, K=2 and M=6, K=2. Figures 7 and 8 illustrate the response of LFC for M=4 and M=6 respectively; while Table 4 summaries the obtained results.

Figure 7. Frequency deviation of third scenario for LFC via Legendre function at M=4, K=2

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Table 4. Results of third scenario for LFC using Legendre function

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As seen from Table 5, we can conclude that the optimal controller via Legendre wavelet function provides better performance and response than LQR. Both controllers offer the same settling time , but the Legendre wavelet function controller gives better performance in terms of the frequency deviation and steady state error by fifteen times less than LQR.

Figure 8. Frequency deviation of third scenario for LFC via Legendre function at M=6, K=2

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Table 5. Comparison for LFC between LQR and Legendre function

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