Improvement of Power Systems Stability Using a New Learning Algorithm Based on Lyapunov Theory for Neural Network

Abstract

In this paper, a new learning algorithm based on Lyapunov stability theory for neural networks is used to improve the power system stability. During the online control process, the identification of system is not necessary, because of learning ability of the proposed controller. One of the proposed controller features is robustness to different operating conditions and disturbances. Moreover, the Prony method is used to obtain the exponential damping of power system oscillations in this paper. The test power systems are a two-area four-machine power system and IEEE 39-bus power system. The simulation results show that the oscillations are satisfactorily damped out by the proposed approach. The proposed approach is effective to mitigate power system oscillations and improve the stability.

Appendices

Appendices

1.1 Appendix A

The parameters used for the conventional PSS (CPSS) [22]:

• Gain: K _p  = 30,

• washout time constant: T _W  = 10.

• Lead-lag #1time constant: T ₁ = 0.05, T ₂ = 0.02.

• Lead-lag #2time constant: T ₁ = 3, T ₂ = 5.4.

1.2 Appendix B

Choose the following candidate of Lyapunov function:

$$V\left( k \right) = \beta^{k} e^{2} \left( k \right),$$

(10)

where β is a positive constant and β > 1. The difference between V(k) and V(k − 1) is then given by

$$\begin{aligned} \Delta V\left( k \right) = V\left( k \right) - V\left( {k - 1} \right) \hfill \\ =\, \beta^{k} e^{2} \left( k \right) - \beta^{k - 1} e^{2} \left( {k - 1} \right) \hfill \\ =\, \beta^{k} \left( {y\left( k \right) - d\left( k \right)} \right)^{2} - \beta^{k - 1} e^{2} \left( {k - 1} \right) \hfill \\ =\, \beta^{k} \left( {\sum\limits_{j = 1}^{n} {w_{1,j}^{(2,1)} \left( k \right)H_{j} \left( k \right)} - d\left( k \right)} \right)^{2} - \beta^{k - 1} e^{2} \left( {k - 1} \right) \hfill \\ \end{aligned}$$

(11)

Using (5) and (6) in (11), we have

$$\begin{aligned} \Delta V\left( k \right) = \, \beta^{k} \left( {\sum\limits_{j = 1}^{n} {w_{1,j}^{(2,1)} \left( k \right)F_{j} \left( {\sum\limits_{i = 1}^{n} {w_{j,i}^{{\left( {1,0} \right)}} \left( k \right)x_{i} \left( k \right)} } \right)} - d\left( k \right)} \right)^{2} - \beta^{k - 1} e^{2} \left( {k - 1} \right) \hfill \\ \, = \beta^{k} \left( {\sum\limits_{j = 1}^{n} {w_{1,j}^{(2,1)} \left( k \right)F_{j} \left( {\sum\limits_{i = 1}^{n} {\frac{1}{{nx_{i} \left( k \right)}}G_{j} \left( {\frac{{\beta^{{\frac{ - k}{2}}} e\left( {k - 1} \right) + d\left( k \right)}}{{nw_{1,j}^{{\left( {2,1} \right)}} \left( k \right)}}} \right)x_{i} \left( k \right)} } \right)} - d\left( k \right)} \right)^{2} - \beta^{k - 1} e^{2} \left( {k - 1} \right) \hfill \\ \, = - \left( {\beta^{k - 1} - 1} \right)e^{2} \left( {k - 1} \right) \prec 0 \hfill \\ \end{aligned}$$

(12)

Therefore, according to Lyapunov stability theory, the tracking error e(k) asymptotically converges to zero.

Appendix C

$$\begin{aligned} e\left( k \right) & = y\left( k \right) - d\left( k \right) \\ & = \sum\limits_{j = 1}^{n} {w_{1,j}^{{\left( {2,1} \right)}} H_{j} \left( k \right) - d\left( k \right)} \\ & = \sum\limits_{j = 1}^{n} {w_{1,j}^{(2,1)} \left( k \right)F_{j} \left( {\sum\limits_{i = 1}^{n} {\frac{1}{{nx_{i} \left( k \right)}}G_{j} \left( {\frac{{\beta^{{\frac{ - k}{2}}} e\left( {k - 1} \right) + d\left( k \right)}}{{nw_{1,j}^{{\left( {2,1} \right)}} \left( k \right)}}} \right)x_{i} \left( k \right)} } \right)} - d\left( k \right) \\ & = \beta^{{ - \frac{k}{2}}} e\left( {k - 1} \right) \\ \end{aligned}$$

(13)

It is noted that

$$\begin{aligned} e\left( 1 \right) = \beta^{{ - \frac{1}{2}}} e\left( 0 \right) \hfill \\ e\left( 2 \right) = \beta^{{ - \frac{2}{2}}} e\left( 1 \right) = \beta^{{ - \frac{1 + 2}{2}}} e\left( 0 \right) \hfill \\ e\left( 3 \right) = \beta^{{ - \frac{3}{2}}} e\left( 2 \right) = \beta^{{ - \frac{1 + 2 + 3}{2}}} e\left( 0 \right) \hfill \\ \vdots \hfill \\ e\left( k \right) = \beta^{{ - \frac{k}{2}}} e\left( {k - 1} \right) = \beta^{{ - \frac{1 + 2 + 3 + \cdots + k}{2}}} e\left( 0 \right) \hfill \\ \, = \beta^{{ - \frac{{\left( {1 + k} \right)k}}{4}}} e\left( 0 \right). \hfill \\ \end{aligned}$$

(14)